Here is my problem, I used the fact that $W(x)=\ln(x)-\ln(W(x))$, replacing $W(x)$ by $\ln(x)-\ln(... $ a lot amount of times and it seems to works for simple $x$ but when I try with, for example, $\ln(-2)/2$ or $i*pi/2+\ln(2)/2$ ,as you like, it doesn't work anymore.
To help you understand the situation I'm in I try to solve $a^b=b^a$ for $a<0$ which means solving $\ln(a)/a=\ln(b)/b$ and as you know $W(-\ln(b)/b)=-\ln(b)$. I can get the result with matlab but most of the time I can't use my computer so I use a Texas Instrument Ti 82 Calculator, I programmed it to calculate real (from $-1/e$ to $+\infty$) Lambert $W$ values and now I try to do it with complex values so i need an algorithmic way to do it.
since no one seemed to be able to help i kept searching and found an answer, using the infinite tower of complex number's propriety : $z^{z^{z^{z^{.^{.^{.^{}}}}}}}=W(-ln(x))/(-ln(x))$.
So with that i can found $W_{0}(x)$ but i have no clue for how to find the complex $W_{-1}(x)$