Both of these spellings exist and are very often used in the wrong context; this can, of course, have a negative impact on the writer/company using the wrong version and can damage hard-won image. Some classic examples of wrong use are outlined below.
This always has the meaning of ‘completing a set/making up a whole’. This is used when saying that a tie goes well with (complements) a shirt, for example. It is also used to describe a number of people making up a group: ‘This ship has a complement of 50.’
This has two meanings: one is that someone is expressing nice things about you, saying that you or something about you looks nice (a compliment/to compliment someone); the other (when used as ‘complimentary’) means that something is free of charge or done as an act of courtesy.
Therefore, it would be wrong to write: ‘The wine compliments the meal.’ This means that the wine tells the meal how nice it is!
What should be written is: ‘The wine complements the meal.’