Not, strictly speaking, a proof, Euler's relation `e^{pi i} + 1 = 0` is a definition allowing one to take a complex power of e. It is, however, consistent with the analytical continuation of exp as defined on the real line. (In other words, it makes excellent sense, and behaves just as one would like).

One of the 2 formulae I scrawled on my helmet in the Army (the other was E=mc^2, not Born to Kill).