  A right quaternion matrix polynomial is an expression of the form
$P(\lambda)= \displaystyle \sum_{i=0}^{m}A_i \lambda^i$, where $A_i$'s are $n
\times n$ quaternion matrices with $A_m \neq 0$. The aim of this manuscript is
to determine the location of right eigenvalues of $P(\lambda)$ relative to
certain subsets of the set of quaternions. In particular, we extend the notion
of (hyper)stability of complex matrix polynomials to quaternion matrix
polynomials and obtain location of right eigenvalues of $P(\lambda)$ using the
following methods: $(1)$ we give a relation between (hyper)stability of a
quaternion matrix polynomial and its complex adjoint matrix polynomial, $(2)$
we prove that $P(\lambda)$ is stable with respect to an open (closed) ball in
the set of quaternions, centered at a complex number if and only if it is
stable with respect to its intersection with the set of complex numbers and
$(3)$ as a consequence of $(1)$ and $(2)$, we prove that right eigenvalues of
$P(\lambda)$ lie between two concentric balls of specific radii in the set of
quaternions centered at the origin. We identify classes of quaternion matrix
polynomials for which stability and hyperstability are equivalent. We finally
deduce hyperstability of certain univariate quaternion matrix polynomials via
stability of certain multivariate quaternion matrix polynomials.
