I think I was a bit loose in the wording in order to try to get the phrasing as compact as I could. The exponential function is continuous proportional growth - the limit of adding a part of a vector to itself proportional to the current vector and the "step size" as you take the step size to zero. (Real numbers are just the one-dimensional special case of vectors here).

Anyway, my point is that Euler's identity generalizes to any number that encapsulates "sideways" in Euclidean geometry. Any quaternion "x" that is a unit of rotation will satisfy "e^pi*x = -1". You can't get that easily from the power series understanding.