The problem of row-sparse signal reconstruction for complex-valued data with outliers is investigated in this paper. First, we formulate the problem by taking advantage of a sparse weight matrix, which is used to down-weight the outliers. The formulated problem belongs to LASSO-type problems, and such problems can be efficiently solved via cyclic coordinate descent (CCD). We propose an extended CCD algorithm to solve the problem for complex-valued measurements, which requires careful characterization and derivation. Numerical simulation results show that the proposed algorithm is robust against outliers and has a higher empirical probability of exact recovery compared with other tested methods.