A novel quaternion-valued least-mean kurtosis (QLMK) adaptive filtering algorithm is proposed for three- and four-dimensional processes by using the recent generalised Hamilton-real (GHR) calculus. The proposed QLMK algorithm based GHR calculus minimises the negated kurtosis of the error signal as a cost function in the quaternion domain, thus provides an elegant way to solve a trade-off problem between the convergence rate and steady-state error. Moreover, the proposed QLMK algorithm has naturally a robust behaviour for a wide range of noise signals due to its kurtosis-based cost function. Furthermore, the steady-state performance of the proposed QLMK algorithm is analysed to obtain convergence and misadjustment conditions. The comprehensive simulation results on benchmark and real-world problems show that the use of this cost function defined by the quaternion statistics in the proposed QLMK algorithm allows us to process quaternion-valued signals and thus, significantly enhances the performance of the adaptive filter in terms of both the steady-state error and the convergence rate, as compared with the quaternion-valued least-mean-square algorithm based on the recent GHR calculus.