A note on convexity of sections of quaternionic numerical range

The quaternionic numerical range of matrices over the ring of quaternions is not necessarily convex. We prove Toeplitz-Hausdorff like theorem: for any given quaternionic matrix, every section of its quaternionic numerical range is convex. We provide some additional equivalent conditions for the quaternionic numerical range of matrices over quaternions to be convex and prove some numerical radius inequalities.