It seems like verb think licenses two consecutive prepositional phrases [PP] as in [i] and [ii]. I’ve not yet found these constructions: ‘think + PP + Noun phrase’ or ‘think + PP + adjective phrase’. So I wonder if ‘what’ preposed from after ‘the film’ in [iii] would replace PP. Does what really replace PP, or is there any proper account?
Think, unless it's the intransitive 'display cognition' sense of Rodin's The Thinker,
is normally transitive, and its normal object is a complement clause describing the thought.
- He thinks that the earth is flat. (tensed complement with that complementizer)
- He never thought to get a permanent visa. (infinitive complement with Equi)
Nouns are ungrammatical as objects for think, as are pronouns, unless they refer to a clause
- *He thinks the President/her/that one/somebody.
- He thinks it, whether he admits it or not.
When a clause is not available, think can use prepositions and variant complement types
to target specific object types and specific thinking types
- I'm thinking of one of Shakespeare's comedies. ('have in mind')
- I'm thinking of/about selling the motorcycle. ('consider V-ing; plan (on) V-ing')
- I'm thinking my way around the obstacle. ('consider; ponder; analyze')
- I think of him as my uncle. ('consider as')
Of the three example sentences,
(i) She thinks of herself as a poet is a 'consider as' construction.
(ii) People are thinking of her for president is a 'consider V-ing' construction.
(iii) What do you think of the film? is a 'consider as' construction, too.
What do you think of the film? is the question from You think of the film (as/in) what way.