General image fusion aims at integrating important information from multi-source images. However, due to the significant cross-task gap, the respective fusion mechanism varies considerably in practice, resulting in limited performance across subtasks. To handle this problem, we propose a novel task-customized mixture of adapters (TC-MoA) for general image fusion, adaptively prompting various fusion tasks in a unified model. We borrow the insight from the mixture of experts (MoE), taking the experts as efficient tuning adapters to prompt a pre-trained foundation model. The task-specific routing networks customize these adapters to extract task-specific information from different sources with dynamic dominant intensity, performing adaptive visual feature prompt fusion. These adapters are shared across different tasks and constrained by mutual information regularization, ensuring compatibility with different tasks while complementarity for multi-source images. Notably, our TC-MoA controls the dominant intensity bias for different fusion tasks, successfully unifying multiple fusion tasks in a single model. Extensive experiments show that TC-MoA outperforms the competing approaches in learning commonalities while retaining compatibility for general image fusion (multi-modal, multi-exposure, and multi-focus), and also demonstrating striking controllability on more generalization experiments. The code is available at https://github.com/YangSun22/TC-MoA.