Climatic comes from the word climate and refers to the weather. The language gods screwed this one up. These homonyms are very similar in meaning. To complement is to add to or reinforce something. The yellow scarf complements her wardrobe.
Compliment means to flatter.
He complimented her on her lovely new scarf . Here's another dandy. Though I was barely conscious, my conscience was nagging me. A careful writer or speaker treats it as plural. The data support no conclusions as yet.