Knowledge representation learning achieves the automatic completion of knowledge graphs (KGs) by embedding entities into continuous low-dimensional vector space. In knowledge graph completion (KGC) tasks, the inter-dependencies and hierarchical information in KGs have gained attention. Existing methods do not well capture the latent dependencies between all components of entities and relations. To address this, we introduce the mathematical theories of octonion, a more expressive generalized form of complex number and quaternion, and propose a translation-based KGC model with octonion (TransO). TransO models entities as octonion coordinate vectors, relations as the combination of octonion component matrices and coordinate vectors, and uses specific grouping calculation rules to interact between entities and relations. In addition, since hyperbolic Poincaré space in non-Euclidean mathematics can represent hierarchical data more accurately and effectively than traditional Euclidean space, we propose a Poincaré-extended TransO model (PTransO). PTransO transforms octonion coordinate vectors into hyperbolic embeddings by exponential mapping, and integrates the Euclidean-based calculations into hyperbolic space by operations such as Möbius addition and hyperbolic distance. The experimental results of link prediction indicate that TransO outperforms other translation-based models on the WN18 benchmark, and PTransO further achieves state-of-the-art performance in low-dimensional space on the well-established WN18RR and FB15k-237 benchmarks.