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Recently, the augmented complex LMS (ACLMS) algorithm has been proposed for modeling complex-valued signal relationships in which a widely-linear model can be more appropriate. It is not clear, however, how the behavior of ACLMS differs from that of the conventional complex LMS (CCLMS) algorithm. In this paper, we leverage a recently-developed analysis for the complex LMS algorithm to illuminate the performance relationships between the ACLMS and CCLMS algorithms. Our analysis shows that the ACLMS algorithm can potentially achieve a lower steady-state mean-squared error as compared to that of CCLMS, but the convergence speed of ACLMS is slowed in the presence of highly non-circular complex-valued input signals. An adaptive beamforming example indicates the utility of the results.