The biggest offense here is calling something without sleeves a jacket

Never heard of Jdownloader, does it offer any benefits over yt-dlp?

The natural representation would be the transient solution u(t) or i(t). Harmonic solutions are merely a special case, for which it turned out complex numbers were useful (because of the way they can represent rotation). They certainly serve a purpose there, but imo this is not an instance of ‘complex numbers appearing in nature’.

But that is hardly a ‘natural occurence’ of complex numbers - it just turned out that they were useful to represent the special case of harmonic solutions because of their relationship with trig functions.

Julia?