Universal uniqueness criterion?

[Base-Acid Tetration]

Infinity can be imagined as a point on the extended complex plane, which is a sphere.
You can regard infinity as any other point on the complex plane.
The tetrational is not holomorphic (not even continuous) at infinity.
However one can approach infinity from different angles/sectors where the limit may be a single value.

I said that the superlogarithm should be negative infinity at the fixed points, instead of infinity*i or = -infinity*i, because the fixed point is approached via a complex iteration, and then infinite negative iteration of exp, ie iteration of log. (fixed points are repelling in graph of exp) Or leave it an essential singularity if there is no way to complex-iterate exp to get an imaginary number.

I guess its about deforming one initial region into the other initial region

How do we go about doing that? It is not given whether the Abel functions are holomorphic outside of their domains.