Zelig (1983)

In this fictional documentary, a man achieves notoriety for his ability to look and act like anyone he meets. With his unique talent for mimicry, Zelig (Woody Allen) ingratiates himself with people from every sector of society. His chameleon-like skill catches the eye of Eudora Fletcher (Mia Farrow), a doctor who thinks Zelig is in need of serious cognitive analysis. Their relationship moves in a direction that's not often covered in medical textbooks.