This research investigates the global $\mu$-stability of a quaternion-valued Cohen-Grossberg neural network with time-varying delays. The study first introduces the concept of time-varying delays into the neural network model, thereby constructing a more precise model that reflects the dynamic characteristics of neurons. Subsequently, by applying the theorem of homeomorphism, the uniqueness of equilibrium points is discussed, and the sufficient conditions for the existence of a unique equilibrium point in the system are determined. In view of the non-commutativity property of quaternion multiplication, the quaternion-valued neural network system is decomposed into four equivalent real-valued neural network systems. On this basis, by constructing an appropriate Lyapunov function, the sufficient conditions for the global $\mu$-stability of the system are obtained. Finally, through numerical simulation experiments, the global $\mu$-stability of the system is verified, thereby confirming the effectiveness and accuracy of the theoretical conclusions.