Compliment and complement, both as verbs and as nouns, when spoken, sound alike and are nearly identical in their spelling; however, they differ substantially in meaning.
Compliment as a verb is to make a laudatory comment (“The teacher complimented the class on the high quality of its essays.”); or as a noun it is a laudatory comment (“John was pleased by the abundant compliments he received from reviewers of his book.”).
Complement as a verb means to supplement appropriately or adequately (“The complex dessert complemented the main course perfectly.”) As a noun it means an adequate supplement (“The bouquet of flowers was a perfect complement to the tablecloth and chinaware.”).
Both words have a positive connotation, but they are not interchangeable. “The School Board has a full compliment of seven members after the election” and “The general moved to compliment his forces at the front for the first time in months” both illustrate a common misuse of compliment; in both sentences it should be substituted with complement. Apparently the misuse of complement where compliment is intended is much less common or likely.
— Ken Butera