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This paper first reviews least mean modulus-Newton (LMM-Newton) algorithm for complex-domain adaptive filters. The LMM-Newton algorithm is effective in making the convergence of an adaptive filter with a highly correlated input as fast as that for the LMM algorithm with a White & Gaussian filter input. However, the filter convergence for the LMM-Newton algorithm is still much slower than for the LMS algorithm. Then, the paper introduces a generalized error modulus (“p-modulus”) and proposes a new adaptive step-size (ASS) control algorithm to be combined with the LMM-Newton algorithm to further improve the convergence speed. Analysis of the ASS-LMM-Newton algorithm is developed for calculating transient and steady-state behavior. Through experiment with simulations and theoretical calculations of filter convergence, we find that the filter convergence is almost the same for any value of p of “p-modulus.” We demonstrate effectiveness of the proposed ASS-LMM-Newton algorithm, while preserving the robustness of the LMM algorithm against impulsive observation noise. Good agreement between simulated and theoretical convergence validates the analysis.