Why Does Quantum Mechanics Use Complex Numbers?

There's recently some interest stirred up in why does quantum mechanics use complex numbers? There are different ways one can get at that question. You can formulate a character foil to quantum mechanics which uses matrices of quaternions instead of complex numbers. We're still at the stage where we can rotate the jewel this way, rotate it that way and then see this faster, this other facet. This is where the jewel that is quantum theory comes from.