Now I was able to generate the graph! Yessss!!!! 
I give two graphs: the first one is the function as given, the second is its conjugate applied to the conjugate of z, i.e. \( \bar{F(\bar{z})} \), which would be equivalent to "continuity from above" at the branch cut from \( b = \eta \) to \( b = -\infty \) in the extended Kneser/disturbed-Fatou tetrational. The second graph is there to better illustrate the continuation of the progression seen in that post about the complex-base attracting-repelling solution that I posted.
main:
conj-conj:
The scale is -40 to +40 (WOW!) on both axes. Zero is in the center.
Wow. It is soooo close to the attracting regular iteration on the real line... I could see why the regular seems so good now as an extension of tetration, and why it seems like it would create an "analytic" tetrational in the base when concatenated with the Kneser iteration at \( b > \eta \). The singularity/branch point of tetration at base \( \eta \) must be incredibly mild. Weeeeeeeeeeerd -- tetration seems once again to be the weirdest complex function I've ever seen.
I notice something interesting here: the regions of "fractal" structure (where it gets really huge) cross across the imaginary axis. I noticed I was having a heck of a time trying to generate something like this with my continuum sum method. Perhaps that is why. There would have been huge values along the imaginary axis and so the "periodic approximation" I was using didn't work due to the ill behavior.
I give two graphs: the first one is the function as given, the second is its conjugate applied to the conjugate of z, i.e. \( \bar{F(\bar{z})} \), which would be equivalent to "continuity from above" at the branch cut from \( b = \eta \) to \( b = -\infty \) in the extended Kneser/disturbed-Fatou tetrational. The second graph is there to better illustrate the continuation of the progression seen in that post about the complex-base attracting-repelling solution that I posted.main:
conj-conj:
The scale is -40 to +40 (WOW!) on both axes. Zero is in the center.
Wow. It is soooo close to the attracting regular iteration on the real line... I could see why the regular seems so good now as an extension of tetration, and why it seems like it would create an "analytic" tetrational in the base when concatenated with the Kneser iteration at \( b > \eta \). The singularity/branch point of tetration at base \( \eta \) must be incredibly mild. Weeeeeeeeeeerd -- tetration seems once again to be the weirdest complex function I've ever seen.
I notice something interesting here: the regions of "fractal" structure (where it gets really huge) cross across the imaginary axis. I noticed I was having a heck of a time trying to generate something like this with my continuum sum method. Perhaps that is why. There would have been huge values along the imaginary axis and so the "periodic approximation" I was using didn't work due to the ill behavior.