Complex Quantum Wave Can the complex nature of quantum wave arise from the fact that particle is represented as wave packet in spatial frequency and particle's total energy is represented as wave packet in time frequency?
Those wave packets are connected since $E=p^2/2m+V$, but since wave-like particle posses definite location and definite energy, they make sense. Now we have $\cos (\text{phase})+i\centerdot \sin (\text{phase})=e^{i \centerdot \text{phase}}$ and with proper summation two dimensional information could be emerged   into one scalar valued function $\psi (x,t)$.
 A: No, this is not a valid explanation. Pardon my possible simplifications, but I understand the reasoning in the following way: 


*

*$k$ bears independent information

*$\omega$ bears independent information

*We can "store" only one piece of information in the real line, so we need "two folders", which is provided by the two parts of a complex number. 


This is simply not how the whole thing works. The argument is completely detached from what the wavefunction actually is and what it is used for. The "information content" of $\omega$ and $k$ is not generally decomposable into separate $A_1(k), A_2(\omega)$, and it certainly isn't divided up the way you suggest. The full $A(k,\omega)$ is not imposed but determined by the Schrödinger equation.
But to give a simple counterexample to show the invalidity of the argument by it's own means: Actual wavefunctions occur in 3D space, so we have $\vec{k}=(k_x,k_y,k_z)$. With $\omega$ this means "four information folders". So complex numbers are not enough. You could pass to quaternions. But we do not do that in quantum mechanics. Why? Because it is not actually needed, the premises of the argument are incorrect.

Physics sometimes just postulates objects and they work - the postulation cannot be fully explained. Nevertheless, I like the argument for the complex wave-function by Jerry Schirmer: We "need" the formalism to produce actually observed wave phenomena such as interference, but also "flat" and "non-wavy" probability distributions, which however have the potential of interfering.
