complement vs. compliment
To complement is to complete something, supplement it, enhance it, or bring it to perfection. For example, your shoes may complement your dress, you and your spouse may complement each other, or minced garlic may complement a pasta dish. To compliment is to give praise. For example, if I were to say that you have a very nice turtle, this would be a compliment to both you and your turtle. Both words also work as nouns whose meanings are easily inferred from the verb senses.
A corresponding distinction applies to the adjectives complementary and complimentary. Complementary things complete, supplement, or bring to perfection. Complimentary means laudatory, and it also has a second, tangential sense: given free or as a courtesy. The coffee in the hotel lobby, for instance, is complimentary. As a verb, compliment occasionally appears in the corresponding sense to give free.
Complement has a secondary, seldom-used noun sense: a full crew of personnel, especially on a ship. (from the grammarist.com)