Tetration of complex bases to complex towers

[andydude]

... the kidneybean ...

lol, I call it the Shell-Thron region, or "the shell" for short, which oddly enough describes both the author and the shape.

So how about the base \( b = 3 + i \). The intuitive/natural Abel function of exponentials associated with this base is:
\( \text{slog}_b(x) = -1 + (0.984001 + 0.124042 i)z + (0.246854 - 0.0364779 i)z^2 + \cdots \)
when expanded at 0. It seems to be fairly close to being real-valued for arguments around 0 to 1.

I have attached 2 plots, the critical function above, and the piecewise extension to all R. Blue is the real part and red is imaginary part.

Andrew Robbins