Visible-thermal Person Re-identification (VT Re-ID) is a challenging task in all-weather surveillance system. Existing methods concentrate on extracting the modality-shared features, ignoring the discriminative inter-modality complementary features. To tackle this issue, we propose a multi-branch modality residual complementary learning method which consists of the modality residual complementary learning (MRCL) module and the multi-branch feature learning (MBFL) module. The MRCL module can be easily integrated into existing CNN baselines and drive the network to focus on both intra-modality and inter-modality information. On one hand, we adopt the basic two-stream network to obtain the intra-modality features, on the other hand, we capture the inter-modality complementary features within the residual image obtained by cross-modality correlation saliency erasing operation. To handle the intra-modality variations, we employ the MBFL module to capture local spatial features and local channel features, then integrate them with global features to achieve part-to-part and high-level semantic information matching. Finally, the discriminability and robustness of the ultimate representations are enhanced by multi-branch constraint loss learning. Extensive experiments on RegDB and SYSU-MM01 datasets demonstrate the superiority of our proposed method compared with state-of-the-art methods.