While the coordinate complexification approach offers several advantages, such as avoiding volumetric modifications and maintaining the integrity of the boundary integral formulation, it does have limitations compared to methods like perfectly matched layers (PMLs) or absorbing regions. One potential drawback is the requirement for careful contour selection in the complex plane. The success of the complexification heavily relies on the choice of contours that ensure the density decays exponentially, which can be challenging in practice, especially for complex geometries or boundary conditions.
Additionally, the coordinate complexification method may not be as effective in scenarios where the outgoing nature of the solution is not well-defined, such as in the presence of multiple scattering events or when dealing with non-linear problems. In contrast, PMLs are designed to absorb outgoing waves more universally, making them more robust in a wider range of scenarios. Furthermore, the implementation of the complexification scheme may require more intricate mathematical analysis and numerical techniques to ensure stability and convergence, which could complicate its application in practical settings compared to the more straightforward application of PMLs or absorbing boundary conditions.