Let
F
\mathbb {F}
denote either the complex numbers
C
\mathbb {C}
or the quaternions
H
\mathbb {H}
. Let
H
F
n
\mathbf {H}_{\mathbb {F}}^n
denote the
n
n
-dimensional hyperbolic space over
F
\mathbb {F}
. We obtain algebraic criteria to classify the isometries of
H
F
n
\mathbf {H}_{\mathbb {F}}^n
. This generalizes the work in Geom. Dedicata 157 (2012), 23–39 and Proc. Amer. Math. Soc. 141 (2013), 1017–1027, to isometries of arbitrary dimensional quaternionic hyperbolic space. As a corollary, a characterization of isometries of
H
C
n
\mathbf {H}_{\mathbb {C}}^n
is also obtained.