Finding meaningful 1-1 correspondences between hippocampal (HP) surfaces is an important but difficult problem in computational anatomy. Unless high-field imaging is used, there are no well-defined anatomical features on the HP that can be used as landmark constraints, so defining meaningful registrations between HP surfaces is challenging. Here we developed a new algorithm to automatically register HP surfaces with complete geometric matching, avoiding the need to manually label landmark features. A good registration depends on a reasonable choice of shape energy that measures the dissimilarity between surfaces. In our algorithm, we first propose a complete shape index using the Beltrami coefficient and curvatures, which measures subtle local differences. The proposed shape energy is zero if and only if two shapes are identical up to a rigid motion. We then seek the best surface registration by minimizing the shape energy. We propose a simple representation of surface diffeomorphisms using Beltrami coefficients, which simplifies the optimization process. We then iteratively minimize the shape energy using the Beltrami Holomorphic flow (BHF) method introduced in this paper. Experimental results on 212 HP of normal and diseased (Alzheimer's disease) subjects show our proposed algorithm is effective in registering HP surfaces with complete geometric matching. The proposed shape energy can also capture local shape differences between HP, for shape analysis in Alzheimer's disease, schizophrenia and epilepsy.