In this research, a novel adaptive filtering algorithm is proposed for complex domain signal processing. The proposed algorithm is based on Wirtinger calculus and is called as q-Complex Least Mean Square (q-CLMS) algorithm. The proposed algorithm could be considered as an extension of the q-LMS algorithm for the complex domain. Transient and steady-state analyses of the proposed q-CLMS algorithm are performed and exact analytical expressions for mean analysis, mean square error (MSE), excess mean square error (EMSE), mean square deviation (MSD) and misadjustment are presented. Extensive experiments have been conducted and a good match between the simulation results and theoretical findings is reported. The proposed q-CLMS algorithm is also explored for whitening applications with satisfactory performance. A modification of the proposed q-CLMS algorithm called Enhanced $q$-CLMS (Eq-CLMS) is also proposed. The Eq-CLMS algorithm eliminates the need for a pre-coded value of the q-parameter thereby automatically adapting to the best value. Extensive experiments are performed on system identification and channel equalization tasks and the proposed algorithm is shown to outperform several benchmark and state-of-the-art approaches namely Complex Least Mean Square (CLMS), Normalized Complex Least Mean Square (NCLMS), Variable Step Size Complex Least Mean Square (VSS-CLMS), Complex FLMS (CFLMS) and Fractional-ordered-CLMS (FoCLMS) algorithms.