For context, I have been relearning a lot of math through the lovely website Brilliant.org. One of their sections covers complex numbers and tries to intuitively introduce Euler's Formula and complex exponentiation by pulling features from polar coordinates, trigonometry, real number exponentiation, and vector space transformations.
While I am now decently familiar with how complex exponentiation behaves (i.e. inducing rotation), I am slightly confused by the following.
$ 2^3 z$ can be viewed as stretching the complex number $z$ by $2^3$. This could be rewritten as $8z$. Therefore, Brilliant.org suggests that exponentiation of real numbers can be thought of as stretching a vector just like real number multiplication would. (check - understood)
Brilliant.org then demonstrates that multiplying $z_1$ by another complex number $z_2$ is equivalent to first stretching $z_1$ by the magnitude of $z_2$ and then rotating $z_1$ by the angle that $z_2$ creates with the real axis counterclockwise. (check - understood)
However, this is where I get confused. Why does, for example, $2^{2i}* z$ cause purely rotation of z but $2i*z$ does not (i.e. it causes stretching, too, in addition to rotation)?
To me, the fact that $2^{(2i+3)}$ causes both rotation and stretching makes perfect sense because we can rewrite this as $(2^3)*(2^{(2i)})$. As previously noted by Brilliant.org, exponentiation by real numbers can thought of as stretching.
Here is the crux of my issue:
I understand that the magnitude of the imaginary number in the exponent (for example, the $'2'$ in $e^{2i}$ ) can be thought of as a rate of speed...but why does this interpretation 'drop' when we are doing something like $2i * z$. i.e. Why is the $2$ in $2i*z$ not also treated like a rate of rotation but instead treated like a magnitude of stretching ?
My math skill is not particularly high level so if anyone can offer as much of an intuitive answer as possible, it would be greatly appreciated!
Edit 1: I guess another way of expressing this question is as follows:
Why does a duality exist between real number exponentiation and real number multiplication but a duality does not exist between imaginary number exponentiation and imaginary number multiplication (i.e. imaginary number multiplication can cause stretching in addition to rotation)?
Edit 2: While I accept that Euler's formula is a way of proving that exponentiation of purely imaginary numbers has a magnitude of 1 and therefore does not invoke stretching, that is not the sort of answer I am looking for. My question is aimed at identifying what was specified in Edit 1.
Edit 3: Here is a picture that helps clarify my point of confusion.
Edit 4: The question that was asked in this post Which general physical transformation to the number space does exponentiation represent? is sort of the theme that I am going for. The answer that was given to this post, however, omits a reference to the complex numbers.