A question concerning uniqueness

[Catullus]

My question is simple, and I hope somebody has an answer because I am a little confused.  How come Tetration has multiple possible extensions to the complex domain that are analytic, but exponentiation only has one?  Is it possible to have an alternative extension for exponentiation that is still analytic?

Why, not how come. Why is better than how come.
Regular exponentiation is not always analytic. For example, is not analytic at zero.
Why a piecewise definition?
Functions defined from piece-wise definitions are typically not analytic at the ends of pieces.
Defining  as might work.
Defining  as  might work.
Defining  as might work.
Though, these are not piece-wise definitions.