Is this a complementizer what?

Question

  He bought Harry a hamburger and they sat down on plastic seats to eat them. Harry kept looking around. Everything looked so strange, somehow.

   "You all right, Harry? Yer very quiet," said Hagrid.
Harry wasn't sure he could explain. He'd just had the best birthday of his life –– and yet –– he chewed his hamburger, trying to find the words.

  "Everyone thinks I'm special," he said at last. "All those people in the Leaky Cauldron, Profes-sor Quirrell, Mr. Ollivander... but I don't know anything about magic at all. How can they expect great things? I'm famous and I can't even remember what (1) I'm famous for. I don't know what happened when Vol-, sorry –– I mean, the night my parents died (2)."
(Harry Potter and the Sorcerer's Stone)

I guess what (1) is a complementizer that moved forth from for's complement. And it’s the pronoun referring to what happened the night my parents died (2), or short form of this. Is this a proper parsing?

Answer 1

"I can't even remember what I'm famous for" contains two clauses.

"What I'm famous for" is a "wh- clause". It can be considered to be the result transformation of the sentence "I'm famous for what" by a process called wh- movement where by "what" moves to the front.

The entire clause is nominalized. That is to say, "what I'm famous for" serves as a big noun.

The pattern "I cannot remember [ ]" requires a noun phrase, or a nominalized clause to fill in the blank.

I do not believe that "what" serves as a complementizer: not in your sentence, or anywhere else.

There is also no evidence of any invisible complementizer in the sentence.

Nominalized clauses can be joined without a complementizer, because they denote something (similarly to a noun) whereas complementized clauses represent themselves as clauses, more or less.

There are wh- words that are complementizers, such: whom, which, whether.

For instance, consider:

The song which you like is playing on the radio.

There is no grammatical way to use "what" in a similar way as a complementizer. "The song what you like" is not grammatical.

And, conversely, "which you like" does not nominalize: "This is which I like" is not grammatical.