Right now I can't think of an example where "complain of" would be correct English. Someone could use the noun form of the word and say "the complaint of", as in "the complaint of your friend is that you study too much". "to complain about" - "about' is the preposition and "complain" is the verb. You can also complain "to" someone "about" something.

"Complain of" is used to describe experiencing pain or discomfort in our bodies, for example, He is complaining of chest pain, She complained of an ear infection.

here is an example for 'complain of' : http://www.thefreedictionary.com/complain

Complain of is used in older English novels. "She is complaining of a slight chill". Modern usage would be "She is complaining about her cold" or "She is complaining that she is sick"