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This paper introduces the concept of complex weighted median (WM) filtering admitting complex weighting. Unlike previous approaches in the literature that only allowed positive real-valued weights, the new WM structures exhibit improved performance as they exploit the richness of unrestricted complex weighting. To this end, the newly defined complex WM structures synthesize filtering operations, whereas the prior structures could only attain smoothing properties due to the inherent constraints imposed on the weights. In order to overcome the two-dimensional (2-D) search burden associated with the computation of the complex WM, two fast, robust, and very efficient approximations are introduced. Adaptive optimization algorithms for their design are developed leading to simple LMS-type weight updates. Several simulations are shown illustrating the performance of the new complex WM filter structures.