What exactly is the phase of a complex number? I've heard the term "phase" several times (e.g. in the context of waves), not just in the context of complex numbers (which are often mentioned e.g. in the context of quantum computing), but, in this post, I would like to know the definition of a phase of a complex number. So, what is it?
Here are a few sub-questions to help you answer
(more thoroughly) the main question above (so you don't have to answer all these sub-questions, but, if you answer them, the main question will probably be answered more satisfactorily).

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*Why is it called a "phase"?

*Why do we care about phases of complex numbers?

*Can a complex number have different phases and still be considered the same complex number?

*What are examples of applications of this property of a complex number?

 A: A complex number can be represented in a "rectangular" or "polar" forms.
The rectangular form of a complex number $z$ is composed of a real part and the imaginary part: $z = a + bi$, where $a$ is the real part and $b$ is the imaginary part, and $i$ is the imaginary unity, $\sqrt{-1}$. We can visualise such number in a 2-dimensional plane , called the "complex plane", as follows:

where "Im" refers to the "imaginary" axis and "Re" to the real axis. Why is it called "rectangular form"? Because of the rectangle we can see in the picture above.
In the polar representation, rather than specifying the coordinates (of the complex plane) $a$ and $b$ of the complex number, we specify the angle (counterclockwise sense) between the real axis ("Re") and the vector representing the complex number, and the length of such vector. Formally, the polar representation of a complex number $z$ is $z=r(\cos \varphi +i\sin \varphi)$, where $\varphi$ is the angle between the real axis and the vector (in blue in the figures) and $r$ is the length (or magnitude or module) of such vector. We similarly visualise such polar representation as follows

where $y=b$ and $x=a$ (that is, we just changed letters). Why is it called "polar form"? Because the angle $\varphi$ is also often called the "polar angle".
Mathematically, the complex number $z=r(\cos \varphi +i\sin \varphi)$ is equivalent to $z=re^{i\varphi}$, because of the Euler's identity, which is $e^{i \varphi}=\cos \varphi + i \sin \varphi$, which holds for any real number $\varphi$.

In our case, the real number $\varphi$ is the angle.
In this context, we often call the phase either the term $e^{i\varphi}$ or simply the angle $\varphi$. Note that the angle $\varphi$ completely determines $e^{i\varphi}$, that is, given $\varphi$, we can easily retrieve $e^{i\varphi}$ (without any other information) by simply replacing this angle in the Euler's identity.

Can a complex number have different phases and still be considered the same complex number?

Suppose that we have complex numbers $z_1 = re^{i2\pi}$ (a complex number of length $r$ which lies on the real axis and points to the right) and $z_2 = re^{i4\pi}$. Clearly, $z_1 = z_2$, but they have different "phases": $z_1$ has phase $e^{i2\pi}$ (or $2\pi$), whereas $z_2$ has phase $e^{i4\pi}$ (or $4\pi$). Hence, $z_1$ and $z_2$, even though they have different phases, they represent the exact same complex number. Hence, in general, two polar representations with different phases can represent the same complex number.

Why is $\varphi$ (or $e^{i\varphi}$) called a "phase"?

Because it refers to the angle around the origin of the complex plane.

What are examples of applications of this property of a complex number?

For example, Vandermonde matrix, which represents the discrete Fourier transform (DFT), is a special matrix. If we look at the Wikipedia article on the DFT matrix, we can see the entries of such vector are complex numbers with different phases.
(Images in this answer are screenshots of the images of Wikipedia articles).
