Is this a complex sentence or compound-complex sentence?
Researchers who study the trajectory of biodiversity loss are alarmed that, within the century, an exponentially rising extinction rate might easily wipe out most of the species still surviving at the present time.
Sentence structure is often simplified into simple, compound and complex structures. Most grammar books share a similar definition: a simple sentence contains a single clause (a clause has only one main verb) e.g. "I like pizza"; a compound sentence contains two or more independent clauses (an independent clause can 'stand alone') and is often linked with FANBOYS (for, and, nor, but, or, yet & so) or BOAS (but, or, and & so) e.g. "I like pizza but I don't like pasta"; a complex sentence contains two or more clauses where there is one independent clause and one (or more) dependent clause (a dependent clause cannot 'stand alone') e.g. "I like pizza because it's delicious". It is also possible to construct compound-complex sentences e.g. "I like pizza because it's delicious but I don't like pasta because it sticks to my teeth and it is usually made with tomatoes, which I don't like".
These definitions are based on 'traditional' grammar and while these definitions are correct most of the time, issues can arise. The most common problem relates to problems of understanding how language functions in context. A systemic functional linguistics (SFL) approach to grammar also introduces a scale of 'rank' when discussing lexicogrammar (so-called because words and grammar are two ends of a spectrum at grammatical level). A rank scale identifies that words make up groups of words, groups of words make up clauses and clauses make up sentences. This can be viewed from both bottom-up (x makes up y, y makes up z etc.) and top-down (z is made up of y, y is made of of x etc.). This is important in your example because clauses are functioning as parts of groups (e.g. a defining clause functions as parts of the nominal group rather than being another clause).
A second major issue is that verbs can function in different ways and have different structures. Traditional grammar identifies transitive (needs an object) and intransitive (doesn't need an object). However, some verbs are followed by "that" clauses e.g. said/know/argue that + clause etc. In SFL these verbs are said to "project: another clause. Another issue is that meaning can be enhanced through the use of 'that-clauses" e.g. so + adj + that, such + noun + that. (See http://www.alvinleong.info/sfg/sfgcomplex.html for more information.)
Looking at your example "Researchers who study the trajectory of biodiversity loss are alarmed that, within the century, an exponentially rising extinction rate might easily wipe out most of the species still surviving at the present time." we can identify a subject, which includes a defining clause (an embedded clause functioning to post-modify researchers), and a main verb are. Following this, a quality, "alarmed" attributes a feeling to the subject - researchers. The attribute is enhanced by adding a 'that clause', which functions to explain the reason why the researchers are "alarmed". As such, although there are a number of clauses in the sentence, there is only one main clause - the researchers are alarmed. The 'subject' is post-modified by a defining clause and the attribute is enhanced by a that-clause which gives a reason.
Following this analysis, I would consider this clause to be a complex sentence. I am aware, however, that there are other ways to analyse grammar and welcome any discussion.
It's a complex sentence.
A quick look at a quick-and-easy resource grammar.about.com produces the definition of a compound-complex sentence:
In English grammar, a compound-complex sentence is a sentence with two or more independent clauses and at least one dependent clause.
The example sentence has only one independent clause, Researchers ... are alarmed - and a couple of subordinate clauses: who study the treajectory of biodiversity loss and that...and exponentially rising extinction rate might easily wipe out most of the species.
It's certainly a complex sentence
a sentence that contains an independent clause and at least one dependent clause.
