Is Lambert W Function in complex domain $(x+iy)e^{x+iy}=a+ib$ solvable? I am solving by method of approximation complex Lambert $W$ Function $(x+iy)e^{x+iy}=a+ib$ for $x$ and $y$, when real values of $a$ and $b$ are given and $i=\sqrt{-1}$. I want to know whether such type of function has been solved earlier. If answer is in affirmative, I pray, I may kindly be informed of its reference or any other source.

To stay with Lambert function.
The equations to be solved are
$$-a-e^x y \sin (y)+e^x x \cos (y)=0 \tag 1$$
$$-b+e^x x \sin (y)+e^x y \cos (y)=0 \tag 2$$
Using $(1)$
$$x=W\left(a e^{-y \tan (y)} \sec (y)\right)+y \tan (y)\tag 3$$
Plug in $(2)$ and you need to solve numerically for $y$
$$\frac{a y \sec ^2(y)}{W\left(a e^{-y \tan (y)} \sec (y)\right)}+a \tan (y)-b=0 \tag 4$$
Trying with $a=\pi$ and $b=e$, Newton method gives
$$y=0.394401045253167$$ from which
$$x=1.194608982218346$$
Checking, the lhs is
$$3.14159265358979 + 2.71828182845905\,i$$
Edit
Discarding Lambert function  and using
$$x=\frac{a \cos (y)+b \sin (y)}{b \cos (y)-a \sin (y)}\,y$$ we need to find the zero of function
$$f(y)=\frac{y}{b \cos (y)-a \sin (y)} \exp\Bigg[\frac{a \cos (y)+b \sin (y)}{b \cos (y)-a \sin (y)} \,y\Bigg]-1\tag 5$$   which will have a solution between $0$ and $\tan ^{-1}\left(\frac{b}{a}\right)$.
Because of this a priori bkown upper bound, it could be interesting to let $y=2 \tan ^{-1}(t)$ and to look for the zero of
$$G(t)={2 \left(t^2+1\right) \tan ^{-1}(t)}\,\,\exp\Bigg[2\frac{at^2-2bt-a }{bt^2+2at-b }\tan ^{-1}(t)\Bigg]+(b t^2+2 a t-b)\tag 6$$
and the solution will easily be bracketed before starting an iterative process.
$$\min \Bigg[0,\tan \left(\frac{1}{2} \tan
   ^{-1}\left(\frac{b}{a}\right)\right)\Bigg] < t_* <\max \Bigg[0,\tan \left(\frac{1}{2} \tan
   ^{-1}\left(\frac{b}{a}\right)\right)\Bigg] $$
