As far I can follow it, it seems that Schroedinger's Equation uses the complex plane in a sense to allow the spatial directions to exist on the real axes (reflecting real space) and put the probability amplitude of the wavefunction on an conveniently "invented" imaginary axis - sort of an additional hyper dimension to house the amplitude. To turn this complex number then into a real probability the Psi wavefunction is squared. I'd like to know why this result isn't then required to be square rooted, to get the correct real number probability magnitude? For example. For two possible states with probability amplitudes summing to 1 on the imaginary axis, they might have the following complex values; 2i/3 and i/3 squaring and taking the moduli gives the following real numbers 4/9 and 1/9 which don't sum to 1, but will if square rooted. Why is this? I think it has something to do with the Normalization process, where the integral of the Psi^2 between - infinty and +infinity must equal 1. But I don't understand why this is Psi^2 and not just the integral of Psi between + & - infinity.