The complex-valued nature of Polarimetric SAR data requires dedicated algorithms that can deal with complex-valued representations. This approach needs to be studied more in the deep learning community, where several works instead transformed the complex-valued signals into the real domain before applying standard real-valued algorithms. In this paper, we employ complex-valued neural networks and study the performance of complex-valued convolutional autoencoders. We demonstrate the ability of such networks to compress fully polarimetric SAR data and decompress them by preserving critical physical properties as revealed by the Pauli and Krogager coherent decompositions and the non-coherent $H-\alpha$ decomposition.