Does anyone know a deeper reason why the quantum mechanical probability amplitude has to be complex? Is it to incorperate time dependence? Or maybe the operator/eigenvector formulation is special and since it includes the scalar product, having complex variables is more general and neccessary? Or maybe the fact that there is some spin with its transformation means that amplitudes should be complex? Hope someone understands these vague ideas No Schrödinger wave argument please, because that doesn't account for the real need of complex numbers.