complement
A complement is something that makes up a satisfying whole with something else. Those shiny red shoes you just bought complement your shiny red purse.
compliment
A compliment is an expression of respect or admiration, like, "You're such a good writer," or "You know so many words!" Keep them coming.
I always admire people who can learn English well. We have so many confusing spellings and words like the different ways to spell pore/pour/poor for example, but all said the same way. It really is a mongrel language.