Objective. Due to non-invasive imaging and the multimodality of magnetic resonance imaging (MRI) images, MRI-based multi-modal brain tumor segmentation (MBTS) studies have attracted more and more attention in recent years. With the great success of convolutional neural networks in various computer vision tasks, lots of MBTS models have been proposed to address the technical challenges of MBTS. However, the problem of limited data collection usually exists in MBTS tasks, making existing studies typically have difficulty in fully exploring the multi-modal MRI images to mine complementary information among different modalities. Approach. We propose a novel quaternion mutual learning strategy (QMLS), which consists of a voxel-wise lesion knowledge mutual learning mechanism (VLKML mechanism) and a quaternion multi-modal feature learning module (QMFL module). Specifically, the VLKML mechanism allows the networks to converge to a robust minimum so that aggressive data augmentation techniques can be applied to expand the limited data fully. In particular, the quaternion-valued QMFL module treats different modalities as components of quaternions to sufficiently learn complementary information among different modalities on the hypercomplex domain while significantly reducing the number of parameters by about 75%. Main results. Extensive experiments on the dataset BraTS 2020 and BraTS 2019 indicate that QMLS achieves superior results to current popular methods with less computational cost. Significance. We propose a novel algorithm for brain tumor segmentation task that achieves better performance with fewer parameters, which helps the clinical application of automatic brain tumor segmentation.