[ameliaquining]

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.

[parsimo2010]

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.

[dhosek]

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.