I thought you just would calculate the base from the fixed point via \(b=z^{\frac{1}{z}}\) and then apply \(\text{LambertW}(-\log(b))/(-\log(b))\) with the right branch?
When I made the animated picture of the primary fixed points, I used these branches \(k\) of the LambertW:
For the attracting fixed point \(z\) in the upper half plane excluding the real line (left of \(e\)) use \(k=1\). Otherwise use \(k=-1\).
For example for \(z_1=2\) is \(b=\sqrt{2}\) then \(z_2 = \text{LambertW}_{-1}(-\log(b))/(-\log(b)) = 4\)
When I made the animated picture of the primary fixed points, I used these branches \(k\) of the LambertW:
For the attracting fixed point \(z\) in the upper half plane excluding the real line (left of \(e\)) use \(k=1\). Otherwise use \(k=-1\).
For example for \(z_1=2\) is \(b=\sqrt{2}\) then \(z_2 = \text{LambertW}_{-1}(-\log(b))/(-\log(b)) = 4\)
