Although you mention a “general complex equation in a certain complex region”, I would still be interested if you are motivated by a certain subclass with more structure to it.
For example, it often happens that such problems originate as nonlinear eigenvalue problems and it is desired to find all roots in a vertical strip, maybe for (in)stability considerations of some underlying dynamical system. In that case, NEP-PACK, also discussed here, may be interesting.
(I cannot help you with its use, but @jarl or his co-authors mentioned in that thread may be able to.)