If you were ever a fan of the classic television program The Honeymooners, you may recall the episode where Ralph Kramden and Ed Norton are learning to play golf. In one of the series' most memorable scenes, Ralph is confused by the direction to "step up and address the ball." By contrast, Norton confidently picks up a club, steps up to the tee, doffs his hat and pleasantly calls out, "Hellooooo, ball!"
This scene comes to mind whenever I read something such as, "the mashed potatoes complimented the chicken." I picture an anthropomorphic mound of mash turning to the chicken and saying, "My, you smell delicious today."
The words compliment (with an "i") and complement (with an "e") are frequently and understandably confused. As a verb, complement means "add to in a way that improves." For example, "The singer's voice was complemented by the music of a jazz trio." Compliment means "politely congratulate or praise." For example, "The singer complimented the jazz trio on their playing."
If we go back to the chicken and potatoes, the potatoes would complement the chicken. The compliments, meanwhile, would go to the chef.
Source: Oxford Dictionary of Current English and The American Heritage Dictionary of the English Language