I've seen this a while ago, and while it's pretty instructive, it's actually also pretty confusing. The magic happens in a seemingly-innocuous throwaway sentence at around 4:20 (after being introduced to the 2D plane):
> ... This can now include rotating along with some stretching and shrinking ...
It's entirely non-obvious WHY we should be okay with rotating all of a sudden. The real answer is not super complicated, but it deals with a couple of amazing relationships between exponential functions and trigonometric identities[1]. So really, we don't have to accept "rotating" as some weird new action we can do when moving into the complex plane, we just have to accept that trigonometry is weirdly related to analysis due to some very cool properties.
> ... This can now include rotating along with some stretching and shrinking ...
It's entirely non-obvious WHY we should be okay with rotating all of a sudden. The real answer is not super complicated, but it deals with a couple of amazing relationships between exponential functions and trigonometric identities[1]. So really, we don't have to accept "rotating" as some weird new action we can do when moving into the complex plane, we just have to accept that trigonometry is weirdly related to analysis due to some very cool properties.
[1] https://www.phy.duke.edu/~rgb/Class/phy51/phy51/node15.html