In the U.S., by the time a student starts calculus they will usually have already worked with complex numbers in the context of elementary algebra. It's true, though, that this won't necessarily have included complex exponentiation.
I'm a calculus teacher (at the moment, as I also teach probability and statistics some semesters). Students are aware of the existence of complex numbers from algebra, in the sense that teachers mention that there are guaranteed to be a fixed number of roots of a polynomial, but these roots might be repeated or complex. They hardly do anything with complex numbers outside this, and do not have enough treatment to define e in such a way.
In fact, it is practically assumed that elementary algebra students have not worked with complex numbers to the extent necessary to understand complex exponentials, due to the fact that complex exponentials are not algebraic.
I’m aware my son is an outlier, but I’m rather proud of his working out the square root(s) of i without even having algebra yet (he’s in fifth grade). Last year I taught him how to solve simple linear equations (ax + b = c) and expanded that to ax + b = cx + d, but he’s been mostly an autodidact with his advanced math (consulting youtube videos and books from the library).
But yeah, there seems not to be a lot of assumption of familiarity with complex numbers beyond the basics of their existence and maybe some simple arithmetic on numbers in the form a + bi which other than i² = −1 is just following the usual rules for polynomial arithmetic. I was surprised at how much basic content on complex numbers was included in the first chapter of my graduate text on complex analysis.
