In this article, a class of quaternion-valued master-slave neural networks (NNs) with time-varying delay and parameter uncertainties was first established by conducting the extension from real-valued chaotic NNs to the quaternion field. Then, based on logarithmic quantized output feedback, the quasisynchronization issue of the NNs was investigated via devising a neoteric dynamic event-triggered controller. In virtue of the classical Lyapunov method and a generalized Halanay inequality, not only corresponding synchronization criteria were obtained to realize the quasisynchronization of master-slave NNs but also a precise upper bound was provided. Moreover, Zeno behavior can be eliminated under the presented scheme in this article. The accuracy of the theoretical outcomes was demonstrated by means of Chua's circuit. Ultimately, some experimental results of pragmatic application in image encryption/decryption were exposed to substantiate the feasibility and efficacy of the current algorithm for the proposed quaternion-valued NNs.