When I look for it, the best I can come up with are things related to making calculations simpler. But I would like something deeper. The best thing I could find is this: In the (translated) words of Jacques Hadamard: "the shortest path between two truths in the real domain passes through the complex domain." http://en.wikipedia.org/wiki/Twistor_space So, it looks like that a shorter then shortest path can be found by making an analytic continuation to complex numbers. Given that Penrose like the Plato's allegory of the cave, I suppose that the shortest path is not in the cave, but in the space from where the projection comes from. Or the true shapes are not in the shadows but behind it. I am not sure of these things. What do you people think?