<p>Modeling the correlation information in the complex spectrum for multi-channel noise suppression remains a significant challenge. Inadequate modeling can lead to the loss of phase information, which results in a decline in signal quality. Moreover, accurately estimating statistical information such as the spectrum or masks is difficult, often resulting in residual noise in the processed signals. To address these challenges, we propose a two-stage deep complex gated convolution attention network (DC-GCAN). The proposed DC-GCAN incorporates a complex Conformer and a complex encoder-decoder with a gated attention mechanism, which effectively models and enhances the complex spectrum, improving the exchange of information between the encoder and decoder. Our two-stage strategy consists of a beamforming stage and a post-filtering stage. In the beamforming stage, we combine the DC-GCAN with the minimum variance distortionless response (MVDR) beamformer to perform spatial filtering of multi-channel signals. In the post-filtering stage, another DC-GCAN is used to further reduce residual noise. Experimental results demonstrate that the proposed method outperforms several competing models on both synthetic speech and non-speech datasets. This indicates that the method can effectively enhance the quality and intelligibility of signals, and is also suitable for applications where the target sound is non-speech. Furthermore, ablation studies confirm the effectiveness of each individual module in the proposed architecture.</p>