The hyperbolic space, characterized by a constant negative curvature and exponentially expanding space, aligns well with the structural properties of heterogeneous graphs. However, although heterogeneous graphs inherently possess diverse power-law structures, most hyperbolic heterogeneous graph embedding models rely on a single hyperbolic space. This approach may fail to effectively capture the diverse power-law structures within heterogeneous graphs. To address this limitation, we propose a Metapath-based Hyperbolic Contrastive Learning framework (MHCL), which uses multiple hyperbolic spaces to capture diverse complex structures within heterogeneous graphs. Specifically, by learning each hyperbolic space to describe the distribution of complex structures corresponding to each metapath, it is possible to capture semantic information effectively. Since metapath embeddings represent distinct semantic information, preserving their discriminability is important when aggregating them to obtain node representations. Therefore, we use a contrastive learning approach to optimize MHCL and improve the discriminability of metapath embeddings. In particular, our contrastive learning method minimizes the distance between embeddings of the same metapath and maximizes the distance between those of different metapaths in hyperbolic space, thereby improving the separability of metapath embeddings with distinct semantic information. We conduct comprehensive experiments to evaluate the effectiveness of MHCL. The experimental results demonstrate that MHCL outperforms state-of-the-art baselines in various graph machine learning tasks, effectively capturing the complex structures of heterogeneous graphs.