Holomorphy for functions with quaternion variables.

Question

Why we cannot define the notion of holomorphy for functions with quaternion variables ?

Answer 1

This is the wrong and outdated answer. Theory of quaternionic differentiability and quaternionic holomorphic functions can be built on principles fully similar (essentially adequate) to ones of complex holomorphic functions. We refuse to consider only the left or only the right approach (regarding another as equivalent) when defining a quaternionic derivative. The left and the right derivatives should be considered only together. It follows that **quaternionic holomorphic functions are those quaternionic functions (in quternon space), whose the left and right derivatives become equal after the transition to 3D space see theory principles here. The quaternionic multiplication of so defined holomorphic functions behave as commutative. The function f(q)=q^2 is quaternionic holomorphic.