The field of digital holography has been significant developed in recent decades, however, the commercialization of digital holograms is still hindered by the issue of large data sizes. Due to the complex signal characteristics of digital holograms, which are of interferometric nature, traditional codecs are not able to provide satisfactory coding efficiency. Furthermore, in a typical coding scenario, the hologram is encoded and then decoded, leading to a numerical reconstruction via a light wave propagation model. While previous researches have mainly focused on the quality of the decoded hologram, it is the numerical reconstruction that is visible to the viewer, and thus, its quality must also be taken into consideration when designing a coding solution. In this study, the coding performances of existing compression standards, JPEG2000 and HEVC-Intra, are evaluated on a set of digital holograms, then the limitations of these standards are analyzed. Subsequently, we propose a deep learning-based compression network for full-complex holograms that demonstrates superior coding performance when compared to the latest standard codecs such as VVC and JPEG-XL, in addition to JPEG2000 and HEVC. The proposed network incorporates not only the quality of the decoded hologram, but also the quality of the numerical reconstruction as distortion costs for network training. The experimental results validate that the proposed network provides superior objective coding efficiency and better visual quality compared to the existing methods.