I’ll admit that this one trips me up, even with 22 years of copy editing behind me. Complimentary, with an i, means either someone is praising you or something is free. It can mean either thing. “She was quite complimentary of your performance.” “The complimentary tickets will be at the ticket stand.”
Complementary, with an e, describes something that enhances or completes something else. “Those turquoise napkins really complement the yellow dishes.” (We can talk later about if we like yellow and turquoise together.)
Here’s my trick: I like to get compliments, and I like free things. Complimentary = i.
Here’s the other trick: You complete me. That enhances things. Complementary = e.