How to find a Newton-like approximation for that function? I want to find the complex fixpoint $t=b^t $ for real bases $b> \eta = \exp(\exp(-1))$.   
added remark: I'm aware that there is a solution using branches of the Lambert-W-function, but I've no Maple/Mathematica and only a rough implementation in Pari/GP for real values. That motivated to try a solution via Newton/Raphson. And to understand and solve such an implementation (which has to deal with derivatives and complex values) is/was then my question here. See also my comment to Fabian's answer below. I seem to have solved it myself for the "principal branch" (see my own answer below) but it is still open for the general case of k'th branch.  [end of remark]  
What I have is a function depending on a parameter $ \beta $ giving the auxiliary values
$$  u = \frac{\beta}{ \sin(\beta) } *\exp( i * \beta) $$
$$   t=\exp(u) $$
$$   b= f(\beta) = \exp(u/t) = \exp(u * \exp(-u))  $$
By this I can do an approximation given a base $B$ using binary search. I can find the bounds of an interval taking lower and upper-limit beta's $ \beta_l = \epsilon $ and $ \beta_u = \pi-\epsilon $ with small epsilons giving the lower and upper bases $b_l$ and $b_u$ respectively. Then comparing $b_m = f(\beta_m)$ where $ \beta_m = (\beta_l + \beta_u)/2 $ with my given base $B$ I can implement a binary search which approximates $b_m$ to $B$ arbitrarily well and having $\beta_m$ I can reconstruct u and the fixpoint t by the above formula.     
However, that binary search needs surprisingly many iterations and I thought, possibly a Newton-like method for that approximation would be more efficient. But since I have complex values involved I do not even see the derivative and even less the formula how to involve that derivative in such an approximation-formula and how to apply this finally to actually do the iterations...
[update 4] moved my own findings into an own answer (as suggested in meta.***)    
[update 1]  (in this old plot I used the letter s instead of b)  
 A: It might be advantageous to reduce the problem to something which has been studied in the literature: the keyword is Lambert W function. The Lambert W function $W(z)$ is defined via $$z= W(z)e^{W(z)}.$$ 
On many systems there are already implementations of this function. In terms of the Lambert W function your $t$ is given via the relation $$t = - \frac{W(-\log b)}{\log b}.$$
We need to find a way to calculate $w=W(z)$ for $z<-1/e$. I follow an idea of this page. Let $z=-r$ (with $r>1/e$), $w=R e^{i \theta}$ ($R>0$). Writing $w e^{w} =z$ in polar coordinates, reduces to the set of equation $$r= R e^{R \cos \theta}$$ and $$\pi =  \theta + R \sin \theta.$$ This set of equations can be solved for $R>0$ and $\theta \in [0,\pi]$. 
The second equation can be rewritten as $R= (\pi - \theta)/\sin \theta$. Plugging this into the first equation yields
$$r \sin \theta= (\pi - \theta) e^{(\pi - \theta) \cot \theta}.$$ This last equation can be solved using bisection (we know that the solution is between 0 and $\pi$). 
A: You are interested in evaluating the function $\exp(-W_k(-\log\,b))$ for various $k$; we can restrict to nonnegative $k$ here due to the conjugate relationship between positive and negative branches. A truncation of formula 86 from this article can serve as a starting point for approximating $W_k(z)$, which can be further polished with Newton-Raphson or Halley:
$$W_k(z)\approx \log\,z+2\pi ik-\log(\log\,z+2\pi ik)-\log\left(1-\frac{\log\,\log\,z}{\log\,z}\right)\left(1+\frac1{\log\,z-\log\,\log\,z}\right)$$
Your course of action, then is to take $z=-\log\,b$, pick your $k$, evaluate the approximation given above, polish with an iterative method, and finally negate and exponentiate.
