Nowadays, adaptive filters in complex domain have received increasing attention since they can be applied to a wider scenario than real valued cases. As two typical information theoretic criteria, the minimum complex kernel risk-sensitive loss (MCKRSL) and maximum complex correntropy criterion (MCCC) have shown robustness to non-Gaussian noises in complex domain adaptive filtering. However, since both criteria cannot consider the probability distribution of error, they are not optimal in the presence of some non-Gaussian noises. This paper firstly defines a complex Shannon entropy in terms of the probability distribution of error. Then, a novel adaptive filter is proposed using the minimum complex Shannon entropy (MCSE) criterion. More significantly, the convergence behavior of the MCSE algorithm is discussed, and thus the steady-state excess mean square error (EMSE) is calculated for theoretical analysis. Finally, simulated results on system identification and channel estimation validate the obtained theoretical results and the merits of the proposed MCSE algorithm.