More Wordstuff

Compliment, with an i, is the word you want when you mean positive commentary, flattery or praise: He complimented her hat, they received many compliments on the show.  In actual usage, the word sometimes takes the metaphorical meaning of improving or reflecting well on (the gold fringe tastefully compliments the red velvet), but strictly it refers to words specifically, actual spoken or written praise, and some readers may still hold this standard strongly enough to make such usage seem jarring or humorous.

Complement, with an e, means something that completes something else, makes a set with it, fills in a missing place or reinforces:  socks and shoes are natural complements to each other, as are hammer and nails, washer and dryer, or even beans and rice.  Very often, complement implies an option, something that could be left out but which enhances or transforms the experience if included; the orange sauce perfectly complements the duck, Mr. Thompson found massive drug use a perfect complement to the Vegas experience.

There are some cases in which either word might be appropriate; a dress could conceivably both compliment and complement someone's figure.  In this instance, though, you'd be better off with a less ambiguous word such as flatter or enhance.

What never works is the belt complimented the shoes, unless you mean the belt can speak, or has gold-foil lettering on it spelling "Hey, check out the hot shoes!"