The complex kurtosis maximization (KM) algorithm is an efficient algorithm for separating mixtures of circular signals and noncircular signals, which are the typical characteristic in real situations. Based on the fixed-point KM algorithm, we here propose a semi-blind complex ICA algorithm by incorporating the magnitude information about a specific signal into the cost function of KM as an inequality constraint. The proposed algorithm is tested using both synthetic signals including circular and noncircular complex-valued sources and real complex-valued functional magnetic resonance imaging (fMRI) data. Performance is compared to several standard complex ICA algorithms and an additional semi-blind complex ICA algorithm based on gradient KM algorithm. The results show that the proposed semi-blind complex ICA algorithm can largely improve the performance of separation. Significant improvement is shown for the detection of task-related components from the complex-valued fMRI data, which are complete but much noisier than the magnitude-only fMRI data.