Why are wavefunctions in Quantum Mechanics shown as complex Circular waves instead of real Planar waves? I'm currently learning Quantum Mechanics from online video lectures and resources. In most of the web articles and videos, the wave functions are shown as circular waves $e^{i\omega t}$ instead of planar waves $\sin{\omega t}$.
[Note: I'm considering a fixed position and hence the equation $e^{i(k\cdot r + \omega t)}$ reduces to $e^{i\omega t}$]
Some examples from the web:
This video shows the wave amplitude to be rotating around the position (i.e. a circular wave in accordance with $e^{i\omega t}$): 
Quantum Wave Function Visualization
The Wikipedia Article on Schrödinger equation describes the plane wave using $e^{i(k\cdot r + \omega t)}$ instead of $\sin{\omega t}$ even though they call it a planar wave:
Schrödinger equation
In this Video the derivation of Probability density is based on a circular wave: Quantum Mechanics 1 Lecture 3
 A: There's a misunderstanding what the word "plane" represents in the term "plane wave". A plane wave is a wave in which the surface of constant phase (wavefront) is a plane:

(image source)
What is shown as a circular thing that rotates for $e^{i\omega t}$ is the phasor that represents the value of the wavefunction at a given (single!) point of space. Phasors are used not only for quantum mechanical wavefunctions: this concept originated in the theory of electric circuits, and is also useful for treatment of other types of waves—even real-valued—e.g. electromagnetic.
What makes quantum mechanical wavefunction special is that it's not usually observable, only its absolute value is. But the effect of interference of quantum particles, like in the double-slit experiment, makes it necessary to introduce an additional parameter to capture this kind of effects. This parameter is the phase, and it's the thing that makes the phasor rotate in the animations you see in the resources on quantum mechanics.
Note that phasor is a vector not in the ordinary physical space: it's a vector in the complex plane, and it doesn't point to any direction in the real physical space, rather being a mathematical abstraction.
A: Basically, the reason we use complex waves in quantum mechanics is because the mathematics is simplest when we do it that way. It is technically possible to formulate quantum mechanics using only real functions, but it is more complicated, and it gives exactly the same results (see the first page of Adler's book Quarternionic Quantum Field Theory for reference: https://projecteuclid.org/download/pdf_1/euclid.cmp/1104115172).
There are many reasons why it is simpler to use complex numbers. For example, differential equations are generally easier to solve using complex waves instead of real waves. Another reason is that the Fourier transform is cleaner with complex waves. The Fourier transform is very important because it relates the "position" of a particle to its "momentum" (although, in general, a single particle will be spread out over many "positions" and "momenta"). Another reason is that the mathematical operations that represent measurements of observables like energy, position, and momentum can be written in a simple form using complex numbers.
A: Remember, that when using rotating phasors to represent out of phase voltages in an AC circuit, only one component of the phasor has physical significance. It does not matter which you choose as long as you keep in mind that that is what you are doing.  If working with phasors in the complex number plane, I would be more comfortable using the real components to represent my physical system.
