A complement clause is a notional sentence or predication that is an argument of a predicate.
The term complement clause is extended by some analysts to include clauses selected by nouns or adjectives.
- I heard the evidence that he did it.
- I am sure that he did it.
- I am not certain what we did.
Relative clauses are not complement clauses. Relative clauses modify a noun phrase, whereas complement clauses are arguments which are selected by a verb, noun, or adjective. In some languages, relative clauses have a gap--a missing NP argument--which is understood to refer to the NP that the relative clause modifies. For instance, in "the person that saw you," the subject of the clause "saw you" is missing, but is understood to be "the person" that the NP as a whole refers to. Complement clauses do not usually have such a gap. For instance, in "the fact that he saw you," the clause "he saw you" does not have any missing arguments. This distinction, however, cannot be used in languages in which it is possible to omit the subject or other clausal arguments freely. This distinction is also not useful in languages which have internally headed relative clauses.
Adverbial clauses are also not complement clauses. Adverbial clauses may modify any verb phrase or sentence, provided they fit semantically, and fill the same role that a purpose, manner, locative or temporal adverb would fill; whereas complement clauses are specifically selected as complements (arguments) by verbs, adjectives or nouns.
We thought that you were coming.
For you to come would be a mistake.
I wonder whether you are coming.
Elsie fled to escape the hurricane. (a purpose clause that is not an argument of a predicate)
Milton came on stage juggling balls. (a manner clause that is not an argument of a predicate)
The mouse ate the cheese that was laying out. (a relative clause that modifies a noun and is not itself an argument of a predicate)
The plumber arrived who we had called earlier. (who we had called earlier is a relative clause—see the discussion)