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.
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)