Infinite tetration of the imaginary unit

[Gottfried]

[update]
The nonconvergent trifurcation of partial evaluation with b=1.8 *i. The actual fixpoint is repelling
   
Data: for the partial evaluation read from left to right, top-down, then this gives the coordinates for
\(
\begin{matrix} {rrr}
1 & b & b^b \\
- & & \\
b^{b^b} & b^{b^{b^b}} & b^{b^{b^{b^b}}} \\
- & & \\
\ldots & \ldots & \ldots \\
\text { } & & \\
& \text{the data are:} & \\
& & \\
1.0000 & 1.8000*I & 0.029026+0.051555*I \\
0.93538+0.071130*I & 0.092461+1.5470*I & 0.045888+0.080842*I \\
0.89836+0.10796*I & 0.13746+1.4245*I & 0.057241+0.10054*I \\
0.87334+0.13111*I & 0.16534+1.3498*I & 0.065447+0.11492*I \\
0.85502+0.14708*I & 0.18474+1.2989*I & 0.071635+0.12595*I \\
0.84094+0.15873*I & 0.19920+1.2619*I & 0.076432+0.13469*I \\
0.82975+0.16753*I & 0.21049+1.2339*I & 0.080221+0.14179*I \\
0.82066+0.17436*I & 0.21959+1.2121*I & 0.083256+0.14765*I \\
0.81314+0.17976*I & 0.22710+1.1946*I & 0.085712+0.15256*I \\
- & & \\
\ldots & \ldots & \ldots \\
- & & \\
trajectory0 & trajectory1 & trajectory2 \\
\end{matrix}
\)


Another better view, comparing 4 different bases:

   

It would be nice to locate two spcial coordinates.

First: what is the value b = x*i, 1.7 < x < 1.75, where convergence suddenly disappears?

Second: what is the value b=x*i, 1.75<x<1.8, where the trajectories give a traight line - if this occurs at all. It seems so, since at x=1.8 the trajectory is continuously left bound and x=1.75 the curve is continuously right bound.


Gottfried