What are the types of complex iteration and tetration?

[JmsNxn]

...complex tetration becomes real tetration when a fixed point at infinity is used.

This is probably absolutely true. And in fact, can probably be derived from Paulsen's paper; and the uniqueness he described.

Essentially if:

\[
F(\pm i \infty) = L^{\pm}\\
\]

And \(F(z) = e^{F(z+1)}\), then \(F(z) = \text{tet}_K(z + z_0)\) for some constant \(z_0\). This should be true. But you may need to tighten your conditions a good amount; particulary that \(\overline{F(\pm i \infty + z)} = F(\mp i \infty + \overline{z})\)--and does so with good asymptotic data. (You would need some knowledge about how it approaches the fixed point at infinity).

PS: Super cool you're sending off a package to Wolfram