Digital filters are employed in hand-held robotic instruments to separate the concomitant involuntary physiological tremor motion from the desired motion of micro-surgeons. Inherent phase-lag in digital filters induces phase distortion (time-lag/delay) into the separated tremor motion and it adversely affects the final tremor compensation. Owing to the necessity of digital filters in hand-held instruments, multi-step prediction of physiological tremor motion is proposed as a solution to counter the induced delay. In this paper, a quaternion variant for extreme learning machines (QELMs) is developed for multi-step prediction of the tremor motion. The learning paradigm of the QELM integrates the identified underlying relationship from 3-D tremor motion in the Hermitian space with the fast learning merits of ELMs theories to predict the tremor motion for a known horizon. Real tremor data acquired from micro-surgeons and novice subjects are employed to validate the QELM for various prediction horizons in-line with the delay induced by the order of digital filters. Prediction inferences underpin that the QELM method elegantly learns the cross-dimensional coupling of the tremor motion with random quaternion neurons and hence obtained significant improvement in prediction performance at all prediction horizons compared with existing methods.