I am not sure where the expression “You complete me” originates. It was certainly popularized by the movie, “Jerry Maguire,” where Tom Cruise’s character declares it to Renée Zellweger. It is the foolish notion that we can be “whole” if we find our “other half,” our “soul mate”—that person who fills our longings. Only God can do that. In fact, He who began His good work within you, promises to complete it (Phil. 1:6).
That's not to say, however, that God doesn't use people in the process. Through personal interactions (that seem to bring out the worst in us), we are made painfully aware of the work God must do in us to fit us together in faith and knowledge until we measure up to the standard of perfection, Christ. (Eph. 4:13).
So as God completes us individually, He fits us into the ‘whole’ until it is perfectly fit for Christ (Eph. 4:16; 5:27; Col. 2:10). So then it is not so terribly wrong to say to one another, “you complete me.”