Naturally complex-valued information or those presented in complex domain are effectively processed by an augmented complex least-mean-square (ACLMS) algorithm. In some applications, the ACLMS algorithm may be too computationally- and memory-intensive to implement. In this paper, a new algorithm, termed partial-update ACLMS (PU-ACLMS) algorithm is proposed, where only a fraction of the coefficient set is selected to update at each iteration. Doing so, two types of partial-update schemes are presented referred to as the sequential and stochastic partial-updates, to reduce computational load and power consumption in the corresponding adaptive filter. The computational cost for full-update PU-ACLMS and its partial-update implementations are discussed. Next, the steady-state mean and mean-square performance of PU-ACLMS for non-circular complex signals are analyzed and closed-form expressions of the steady-state excess mean-square error (EMSE) and mean-square deviation (MSD) are given. Then, employing the weighted energy-conservation relation, the EMSE and MSD learning curves are derived. The simulation results are verified and compared with those of theoretical predictions through numerical examples.