The answer to your question is yes, there are.
In fact, adjectives can be complements in all of their six functions (that I know of):
[1] NON-COMPLEMENT COMPLEMENT
1. attributive That was a legal play. I need a legal adviser.
2. predicative He died young. He was/seemed young.
3. postpositive I need something general. He is the Surgeon General.
4. predeterminer It is such a nuisance. It was such a big problem that
we called the police.
5. fused ? The rich are to blame.
modifier-head
6. subject N/A Big is what you are, my boy.
In [1], in all the entries except in 5, the boldfaced words are adjective phrases (AdjPs). See below for explanations of why they are complements in the right column, but not in the left. In 5, the boldfaced words are the fused modifier-head of a noun phrase (NP).
The adjective will always be a complement in 6 (the subject); it is usually (and perhaps even always) a complement in 5 (fused modifier-head). It is most commonly a complement in 2 (predicative). In 1 and 3 (attributive and postpositive uses), adjectives are by far most commonly modifiers, but sometimes they can be complements, too. Finally, in 4 (predeterminer), they can definitely be either complements or modifiers, but I don't know which role is more common.
Let me say a bit more about 5 and 6.
In 5, the adjective functions as a fused modifier-head of a noun phrase (NP). Such an NP may be either the subject or the object:
[2] a. The rich are to blame. [subject]
b. Let us blame the rich. [object]
Here the rich is a complete noun phrase, even though it has no nouns. It functions as the subject in a. and as the object in b. Both of those functions are complements in the structure of the clause.
In general, NPs can appear in a number of non-complement functions as well. However, I don't know if fused modifier-head NPs ever appear in such roles. I certainly don't have a ready example.
Finally, it is even possible for AdjPs to serve as the subject, as in
[4] Big is what you are, my boy.
This is obviously a complement role, but not one of a predicative complement. Here we have an unusual syntax for a semantic relation that is usually expressed as You are big, where big is a PC. I'll discuss this more below.
Discussion
Consider the following sentences:
[5] a. These are happy people. [attributive]
b. These people seem happy. [predicative complement]
c. I want to meet someone happy. [postpositive]
In all three sentences, we have the same adjective, happy. But in each sentence, that one and the same adjective performs a different function: it is used differently. In [5] a., it is used attributively; in b., as a predicative complement; and in c., postpositively. These are the three principal uses of adjectives. However, there are also some others; we'll get to them later.
Technically, we actually don't want to say that it is the adjectives that perform these roles. Instead, we want to say that
Strictly speaking it is AdjPs [adjective phrases] that occur in these functions, but we can talk of attributive, predicative, and postpositive uses of an adjective with the understanding that the adjective is head of an AdjP in the function in question. The AdjP will consist of the head adjective alone or accompanied by its dependents. (CGEL, p. 529)
In [5], the AdjPs all consist of a single adjective, happy. But if you replace happy by very happy, you get a more complicated AdjP. And this more complicated AdjP nicely functions in all three roles.
Complements vs adjuncts
1. Attributive function
According to CGEL (p. 528),
Attributive AdjPs are almost always modifiers rather than complements.
There are, however, some adjectives that can function attributively as complements. For example (CGEL, p. 439; the number in the original was 2 rather than 6),
[6] ii a. a legal adviser b. an ecological expert
Why are these complements of the noun rather than modifiers? Compare
[7] i a good adviser = an adviser that is good
ii a legal play = a play that is legal
iii a legal adviser ≠ an adviser that is legal
In [7] i, good is a modifier (rather than a complement), and there is a correspondence between attributive and PC functions. Similarly, legal is a modifer in ii, and there is the same correspondence. However, in iii, there is no such correspondence. This happens with only some nouns, and for each such noun, only with some adjectives: for example, with the noun adviser, it happens with such adjectives as legal, technical, medical, etc. These adjectives are therefore licensed by the noun; and whenever something is licensed, it means it is a complement; that, indeed, is the basic criterion (CGEL, p. 440).
Here is another test to distinguish between modifiers and complements among the AdjPs functioning as pre-head dependents in NP structure: the scope of anaphora (CGEL, pp. 440-441).
This test distinguishes between the two senses of the ambiguous
nominal criminal lawyer (with criminal an adjective in both). In the sense "lawyer who works in the field of criminal law", criminal is a complement and resists combination with one: ?I needed a civil lawyer, but he had found me a criminal one. But in the sense "lawyer who is criminal", criminal is a modifier and combines readily with one: It turned out that he was an honest lawyer, not a criminal one, as I'd been led to expect. (CGEL, p. 441.)
Now compare
[8] i I prefer good advisors to bad ones.
ii I prefer legal plays to illegal ones.
iii ?I prefer legal advisors to medical ones.
Here [8] iii is an attempt to combine the pro-form one with an internal complement. Such attempts, according to CGEL, typically show 'varying degrees of infelicity'. In contrast, there is no problem when one is combined with a modifier, as in i and ii.
2. Predicative complement function
As you notied yourself, in this function, an AdjP is always a complement, really by definition. However, there is a similar role where it is an adjunct, a predicative adjunct. We've seen an example of that in [1]:
[1] NON-COMPLEMENT COMPLEMENT
predicative He died young. He seemed young.
Young is a PC in the right column, but a predicative adjunct in the left. I'll return to this below.
3. Postpositive function
According to CGEL (p. 529),
Postpositive adjectives are much less frequent than attributive and predicative ones: adjectives are admissible in this position only under severe syntactic constraints.
Usually, they are modifiers, like happy in [5] c. However, here are some examples of a postpositive adjective functioning as complements (Garner's Modern American Usage, p. 627):
[9] attorney general
surgeon general postmaster general secretary general
ambassador-designate heir apparent
president-elect
accounts payable accounts receivable annuity certain
condition precedent condition subsequent
court-martial knight-errant minister extraordinary
notary public battle royal body politic
First of all, I suppose an argument is required to establish that some of these are indeed adjectives. For example, in the case of general, if it were a noun, the plural would be e.g. *surgeon generals, but it is not: it is surgeons general. Attorney general is an exception because in British English the plural is attorney generals; in American English, however, it's attorneys general (Garner's Modern American Usage, p. 617). I will not try to give a further argument, and just note that Garner's lists all of these as examples of postpositive adjectives.
Secondly, one should provide an argument that these adjectives are complements. This is pretty straightforward: all of these adjectives are licensed. For example, we can use general in this way with only a handful of nouns. The postpositive uses of the other adjectives in [9] are similarly restrictive.
Adjectives in other functions
While the three functions in [5] are the main functions that adjectives (that is, AdjPs) perform, sometimes they can also perform other functions. Here are some examples (CGEL, p. 529; in the original, the example number is 3, not 10):
■ Further adjectival functions
[10] i such a nuisance so serious a problem [predeterminer]
ii the rich the bigger of the two the most useful of them [fused modifier-head]
iii He died young. They served the coffee blindfolded. [ predicative adjunct]
iv Furious, he stormed out of the room. [ predicative adjunct]
Predeterminer AdjPs occur as external modifier in NP structure, preceding the definite article a. This construction is subject to highly restrictive structural conditions described in §3.3. All adjectives that can head a predeterminer AdjP can also be used attributively—cf. such tools, a serious problem.
The fused modifier-head AdjPs in [10ii] combine the functions of internal modifier and head in NP structure; this construction is described in Ch. 5, §9.3. All adjectives that can function in this construction can also be used attributively.
Finally, the AdjPs in [10 iii—iv] function as predicative adjunct. Those in [10 iii] are integrated into clause structure and hence modifiers, while that in [iv] is detached and hence a supplement. All adjectives that can function as predicative adjunct can also function as predicative complement.
An AdjP in the function of a predicative adjunct is obviously not a complement (that's kind of the whole point of this special role).
However, AdjPs functioning as predeterminers can be complements:
4. Predeterminer function
Consider
[11] [How big a company] is it? It was [so serious a matter that we called the police].
Here brackets enclose noun phrases, while the boldfaced words are AdjPs (CGEL, p. 551). It is pretty clear that the AdjPs are complements here; omitting them results in nonsense.
5. The function as the fused modifier-head of an NP
As I've discussed above, fused modifier-head AdjPs are heads of NPs, and these can function as complements in the structure of a clause (i.e. the subject and the object).
[2] a. The rich are to blame. [subject]
b. Let us blame the rich. [object]
6. The subject function: semantics vs syntax
To begin with, consider
[12] I became sick.
As far as syntax, sick functions as a predicative complement (PC). As far as semantics, it denotes a property ('being sick') that is predicated of something (the speaker).
Here the grammatical and syntactic functions align nicely: we accomplish the semantic goal of assigning a property to the speaker (I) by using a complex-transitive verb (become) with a PC complement (sick).
But we don't always have such a straightforward fit between syntax and semantics. Consider the sentence
[13] Big is what you are, my boy.
Semantically, this is similar to [12]: we are predicating a property ('being big') to something (you, 'the boy'). However, syntactically, things are very unusual: consider the word that denotes the property, big; technically, we should say that this is an AdjP consisting of just the head, a single adjective. This AdjP is the subject rather than a PC. And what you are is not the PC here. This we know because is cannot be replaced by any other verb that normally allows a PC (such as become, appear, seem, etc.).
The syntax of [13] is relatively complex, and it would take us too far to analyze it right now. My only point is that the main verb here (is) does not have a PC.1
1Though the second verb, are, does have a PC: compare with Big is what you have become, my boy. However, what you [are]/[have become] is not a clause, but what's called a fused relative noun phrase... like I said, it's complicated.
So, is big in [13] a 'predicate adjective'? Semantically, yes; syntactically, no. And so, syntactically, it is an example of an adjective in complement function which is not a predicative complement. (It should be clear why it is in a complement function: the subject is always a complement.)
