Why is the complex plane shaped like it is? It's always taken for granted that the real number line is perpendicular to multiples of $i$, but why is that? Why isn't $i$ just at some non-90 degree angle to the real number line? Could someone please explain the logic or rationale behind this? It seems self-apparent to me, but I cannot actually see why it is.
Furthermore, why is the real number line even straight? Why does it not bend or curve? I suppose arbitrarily it might be strange to bend it, but why couldn't it bend at 0? Is there a proof showing why?
Of course, these things seem natural to me and make sense, but why does the complex plane have its shape? Is there a detailed proof showing precisely why, or is it just an arbitrary choice some person made many years ago that we choose to accept because it makes sense to us?

Because of complex conjugation, the complex numbers have a natural measure of distance; the "size" (i.e. distance from zero) of a complex number is
$$ \| z \| = \sqrt{z \bar{z}} $$
The right hand side is well-defined in terms of more basic concepts, because $z \bar{z}$ is a nonnegative real number.
More generally, the distance between any two complex numbers is
$$ \text{Distance}(z, w) = \|z - w\| $$
which is enough to reproduce plane geometry. e.g. we can see that $1$ and $i$ are at right angles, because they describe two of the sides of a right triangle with $0$ as the remaining vertex:
$$ \|1\|^2 + \|i\|^2 = \|1-i\|^2 $$

Furthermore, the two main ways of expressing complex numbers can be motivated for purely algebraic reasons. It is convenient to combine conjugation with addition to split a number into two parts
$$ 2z = (z + \bar{z}) + (z - \bar{z}) $$
The first and second terms ultimately lead to the usual real and imaginary parts of $z$.
Combining with multiplication instead suggests, for $z \neq 0$,
$$ z^2 = (z \bar{z}) \cdot \frac{z}{\bar{z}} $$
The first term leads to the norm as described earlier. The second term leads to reducing $z$ to a "unit vector" $\frac{z}{\|z\|}$ on the circle, and thus basically corresponds to the complex argument (i.e. angular position relative to the origin).
