Though it’s probably the most commonly used definition for complex exponentiation, e^i pi=-1 is not an axiom— it is derived from the basic rules of integer arithmetic expanded in scope until they make a closed system.

And from this, we can derive all of the trigonometric functions using only algebra and no geometry.

(See http://www.feynmanlectures.caltech.edu/I_22.html , particularly section 22-5)