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• A novel complex CNN is constructed, in which the operation rules of convolutional layer, subsampling layer, normalization layer, fully-connected layer and activation function are redefined in complex field. • Based on the proposed complex CNN, the multiscale deep Contourlet filter banks are constructed in order to extract robust discriminative features with multidirection, multiscale, and multiresolution properties. • The proposed complex Contourlet-CNN is successfully applied for PolSAR image classification. A complex multiscale network named complex Contourlet convolutional neural network (complex Contourlet-CNN) is proposed for polarimetric synthetic aperture radar (PolSAR) image classification in this paper. In order to make full use of the phase information of PolSAR image, we redefine the conventional operations of CNN in complex field, and the data sets and parameters are always expressed through the complex matrixes in complex CNN. Moreover, the multiscale deep Contourlet filter banks are constructed to extract more robust discriminative features with multidirection, multiscale, and multiresolution properties, which can improve the performance of complex CNN by replacing filters of the first complex convolutional layer with the multiscale deep Contourlet filter banks. The Contourlet transform is used for helping the complex CNN network to capture abstract features in a certain direction and frequency band. Furthermore, the proposed network based on the multiscale geometric properties of Contourlet transform can retrieve the information in the region and direction corresponding to the extracted features. Experiments on different spatial resolutions and land coverings of Flevoland, San Francisco Bay, and Germany PolSAR images show that less training data is required and the performance of the proposed explainable deep learning method is comparable to that of the existing state-of-the-art methods. [ABSTRACT FROM AUTHOR]