Yeah, you can have the same thing with degrees if you use a different exponent base than e: e^((pi/180)ix) = (e^(pi/180))^ix = E^ix and therefore E^1

Yeah, you can have the same thing with degrees if you use a different exponent base than e: e^((pi/180)ix) = (e^(pi/180))^ix = E^ix and therefore E^180i = -1 where E is about 1.0176065. The particular base e has some nice properties, though, like that its rate of change d/dx e^ix at a given point x is just i e^ix. The rage of change for E^ix, on the other hand, is d/dx E^ix = (pi/180) i E^ix, which is a little less "natural". This strange fact -- that using radians makes the expression e^ix = cos x + i sin x have a nice derivative -- is one of the reasons why mathematicians like to define these functions in terms of radians instead of degrees.