You've mixed up compliment and complement. Compliment is something you communicate to someone, such as "You have gorgeous hair" or "You're the greatest!". Complement means that which completes, or adds something different or makes perfect, such as "soda water is the perfect complement to whisky" or "he complements the talents of all the others".
These two trip up almost everybody, because they sound identical and sit only one letter apart. Let me give you a trick that has never failed me. "Complement" with an "e" is about completing, and both words share that "e" in the middle: complete, complement. If something completes or goes well with another thing, reach for the "e" spelling.
"Compliment" with an "i" is the nice thing you say. Think of the "i" in "I like it", or the fact that you give a compliment when you praise. A free gift "with the compliments of the management" is the same word: something given kindly.
So a scarf can complement a coat, while a stranger pays you a compliment on it. The wine complements the meal, and the chef receives your compliments. Get the pair straight and a whole family of words falls into line behind them: complimentary tickets are free, while a complementary colour completes the set.
This is the sort of thing a spellchecker will wave straight through, because both spellings are real words. That is exactly why you have to know it yourself.