One of the derivations of Schroedinger's Equation that I came across assumed the wave function for the particle to be $$\Psi = e^{i(kx-\omega t)} ,$$ and I hardly understand why.  

This equation implies that $$\Psi = cos{(kx-\omega t)}+ isin(kx-\omega t). $$
I came across [this answer](https://physics.stackexchange.com/a/46054/259747) and I believe what it is meant to convey is that the complex expression of the wave function works like a coordinate system that largely simplifies calculations.  

I'm trying to understand the physical meaning of this. Does this mean that the wavefunction has a travelling cosine wave along, say, an $x$-axis and a travelling sine wave along the $y$-axis, and the actual wavefunction is a superposition of these?  

*Sorry if all this sounds absurd. I am an absolute beginner to this course. I'm looking for an intuitive answer more than a mathematical one. I did my research, but it requires some heavy mathematics and I haven't started off with any rigorous math courses at college(I am familiar with basic calculus and linear algebra but not Fourier Transforms or Hilbert Spaces, which often popped up when I looked for explanations.)*