Visible images provide rich details and color information only under well-lighted conditions while infrared images effectively highlight thermal targets under challenging conditions such as low visibility and adverse weather. Infrared-visible image fusion aims to integrate complementary information from infrared and visible images to generate a high-quality fused image. Existing methods exhibit critical limitations such as neglecting color structure information in visible images and performance degradation when processing low-quality color-visible inputs. To address these issues, we propose a quaternion infrared-visible image fusion (QIVIF) framework to generate high-quality fused images completely in the quaternion domain. QIVIF proposes a quaternion low-visibility feature learning model to adaptively extract salient thermal targets and fine-grained texture details from input infrared and visible images respectively under diverse degraded conditions. QIVIF then develops a quaternion adaptive unsharp masking method to adaptively improve high-frequency feature enhancement with balanced illumination. QIVIF further proposes a quaternion hierarchical Bayesian fusion model to integrate infrared saliency and enhanced visible details to obtain high-quality fused images. Extensive experiments across diverse datasets demonstrate that our QIVIF surpasses state-of-the-art methods under challenging low-visibility conditions.