Fixed Point of $x_{n+1}=i^{x_n}$

Question

For $x \in \Bbb C$, let $f(x)=i^x = \exp(i\pi x)$, where $i^2=-1$. Then find the fixed points for $f$.

EDIT: Let for all $n\geq 1$ $$\large a_n=\underbrace{i^{i^{\cdots i}}}_{\text{$n$ times}}$$

My question is, does the sequence of tetrations $\{a_n\}_{n\geq1}$ converge to some complex number? If yes, then what is it?

Answer 1

We had this question quite recently, as I recall. But I cannot now find it. Anyway, here we see $a_n$ for $n$ from $1$ to $50$. They are converging, right?

The limit is: $$ \frac{2i}{\pi}W\left(\frac{-i\pi}{2}\right) \approx .4382829366+.3605924718 i $$ Of course $W$ is the Lambert W function.