Infinite tetration of the imaginary unit

[Gottfried]

Ioannis kindly gave an approximation for the first of the two questions,

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

which seemed to be improvable, so I use that here.

At b~ 1.71290*I there seem to be a limit-case; from the plot it seems, the trajectory connects then the three tri-furcation-branches, which would then give an "aequator"-trajectory (but the current value may not be a good enough approximation to the true limit-case)

   

The points of the three branches do not match, so the question is then, whether in the limit the equator is a smooth or even continuous line...
New question to investigate.

Btw b/2 is 0.85645*I , a number which looks somehow familiar to me, but don't have a clue yet.

Gottfried