as a masters in mathematics, this is exactly what i came here to write. thanks!
To have a notion that multiplication by imaginaries causes rotation, you'd need Euler's formula. I honestly think the best way to get a visual sense for why multiplication by rexp(itheta) is to look at the first few terms of the taylor series added together and see that the adders combine into a spiral that converges on rcos(theta) + isin(theta). that is still plenty visual for me.
To have a notion that multiplication by imaginaries causes rotation, you'd need Euler's formula. I honestly think the best way to get a visual sense for why multiplication by rexp(itheta) is to look at the first few terms of the taylor series added together and see that the adders combine into a spiral that converges on rcos(theta) + isin(theta). that is still plenty visual for me.