It’s definitely one of those cases where you follow the notation and use the rules far beyond the analogies they were based on. Humans discovered exponentiation by thinking about repeated multiplication, so what the heck does it even mean to raise a number to an imaginary power? Well, we have this thing called a Taylor series expansion, let’s try that, oh look, e^ix expands to exactly the sum of the expansions of cosx and isinx, oh look, that’s turns out to be so useful!
“There is something fascinating about science. One gets such wholesale returns of conjecture out of such a trifling investment of fact” - Mark Twain
See also: essentially all of Alice in Wonderland, which almost everyone seems to insist on mistaking for a children’s book, even after they know Lewis Caroll taught mathematics at Oxford.
