This paper investigates spherical regions that contain all the zeros of the quaternion polynomials with quaternion coefficients. By applying Ostrowski-type theorems and leveraging matrix similarity transformations within the quaternion matrix setting, we establish refined bounds for the location of these zeros. The results not only unify and generalize several existing theorems but also yield sharper bounds than those currently available in the literature.

In comparison to quaternions, split quaternions exhibit a more intricate algebraic structure, characterized by the presence of nontrivial zero factors. Furthermore, in various fields such as geometry and electromagnetism, split quaternions serve as more convenient research tools than quaternions. This paper aims to establish the necessary and sufficient conditions for the existence of (anti-) η-Hermitian solutions to a novel system of matrix equations formulated over split quaternions. We also provide the general expression of such solutions when the system is solvable, along with the least squares (anti-) η-Hermitian solution in the case of inconsistency. To illustrate the key findings of this paper, numerical examples are presented.