Question:

Do your shoes compliment or complement your outfit?

Answer:

complement

How to remember it:

If one thing complements (with an "e") another, it completes that thing (e.g., the shoes complete your outfit, or make it perfect). Complement comes from the Latin word for complete.

If you compliment something, you express admiration for it.

And when something is given free as a courtesy or favor, it's complimentary.