I think I have a fairly good grasp of the basics of complex numbers, in particular of how the exponential function is defined.
Usually we define a scalar product on $\mathbb C$ viewed as an $\mathbb R$-vector space, which induces the structure of a normed vector space by means of the 2-norm ${\left \| \cdot \right \|}_2$. However, we might have chosen any other norm.
But to define the exponential function, all we needed was the multiplication structure (and addition) on $\mathbb C$. We never needed to choose the 2-norm! So
why does the set of values $\mathrm{exp}(i\mathbb R)$ coincide with the sphere of the 2-norm, and not any other?
I do see that multiplication by $i$ is an isometry with respect to the $2$ norm, but it also is with respect to the other ones (e.g., if you rotate a square by 90 degrees you still get a square).