Is it necessary for the Imaginary-axis to be perpendicular to the Real-axis? The Real number line is in one dimension. If you want to map a complex number, you would have to add a second dimension to that number line-   the Imaginary-axis.
The Imaginary-axis is always perpendicular to the Real-axis. Here is my question:
Would you still be able to use the complex plane if the imaginary-axis wasn't perpendicular to the Real-axis? In other words, is it still possible to prove theorems involving complex numbers (in a geometrical way) if the Imaginary-axis wasn't perpendicular to the Real-axis?
For example: Could you prove Euler's formula if the Imaginary-axis was tilted to a 45 degree angle?

Sure, you could construct the complex numbers with some other "basis" besides $1$ and $i=\sqrt{-1}$.  What about $1$ and $k={-1}^{1/4}= \frac{1+i}{\sqrt{2}}.$ Addition will still work and you will have some other rule of multiplication:
For example, with $z_1= a_1+ k b_1$ and $z_2=a_2 + k b_2$,
$$\begin{aligned} z_1 z_2 & =(a_1 + k b_1)(a_2 + k b_2) \\& = a_1a_2+k (a_1 b_2+ a_2 b_1)+ k^2 b_1 b_2 \\& = a_1 a_2 + k (a_1b_2 + a_2b_1)+(\sqrt{2}k-1)b_1b_2\\&= 
a_1a_2-b_1b_2+k(a_1b_2+a_2b_1+\sqrt{2}b_1b_2)\end{aligned} $$
So this other "complex number system" is closed under addition and multiplication.
Similarly, we can show that the "polar form" is okay too.
