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?

Complex numbers can be constructed as couple of real numbers :
$
a+ib=(a,b)
$
with suitable definitions of the operations of sum and product ( see here).
With such definitions a complex number corresponds, in a natural way, to an element of $\mathbb{R}^2$ and we have : $1=(1,0)$ and $i=(0,1)$ and , using the usual representation of an orthogonal system for the coordinates in $\mathbb{R}^2$, these two vectors, are represented as points, at distance $=1$ from the origin, on two straight lines that forms a $90°$ angle. So  this same representation is also adopted for the complex numbers. 
