A widely linear quaternion-valued least-mean kurtosis (WL-QLMK) algorithm is introduced for adaptive filtering of quaternion-valued circular and noncircular signals. In the design, kurtosis-based cost function is first defined in the quaternion domain by integrating the widely linear model, and augmented statistics, and then minimized using the recently developed generalized Hamilton-real (GHR) calculus. In this way, the novel WL-QLMK algorithm is obtained for training quaternion-valued adaptive filter structures. Furthermore, its steady-state performance is theoretically analyzed to determine the bounds of the step size, which provides a theoretical justification for simulations. The simulation results over both benchmark system identification scenarios, and one-step-ahead predictions of real-world 4D pathological resting tremors show that the proposed WL-QLMK algorithm, by virtue of its newly defined cost function, significantly enhances the performance compared to the recently developed quaternion-valued algorithms, especially for noncircular signals.