Compliment or Complement?

I am so tired of seeing these mixed up.

Complimentary – Relating to a pleasant remark given to another person
Complimentary – Something given at no charge

“She complimented me on my new haircut.”
“Please accept this wine with our compliments.”
“Frequently complimented by customers for prompt and courteous service.”

Complementary – Things that match or correspond to each other
Complementary – Providing, or serving as, a complement (something that completes)

“The colours in the bathroom complement one another very nicely.”
“Acupuncture is a very effective complementary therapy for back pain.”
“My post-graduate qualification complements my experience as a professional hitman.”

Okay?

Vizzini: “He didn’t fall? In-con-ceiv-able!”
Inigo: “You keep using that word. I do not think it means what you think it means.”