I often see these two adjectives confused.

Problem:
The adjectives “complementary” and “complimentary” are not synonyms.

Explanation:
The adjective “complementary” means forms a complement, and a “complement” is something that completes.

The adjective “complimentary” means expressing or of the nature of a compliment, and a “compliment” is related to the word “comply” and is an expression of praise.

Therefore:

• “John has complementary dishes.” — spelled with an “e” — means that John has dishes that complete or match one another;
• “John has complimentary dishes.” — spelled with an “i” — means that John has dishes that he got as a gift.

A simple way to distinguish “complementary” from “complimentary” is to remember that “complementary” and “complete” both begin with “comple-“.

Solution:
Use “complementary” when you are talking about someone or something that completes someone or something else. Use “complimentary” when you are talking about praise or a gift.