Hyperquaternions being defined as a tensor product of quaternion algebras (or a subalgebra thereof), they constitute Clifford algebras endowed with an associative exterior product providing an efficient mathematical formalism for differential geometry. The paper presents a hyperquaternion formulation of pseudo-euclidean rotations and the Poincaré groups in n dimensions (via dual hyperquaternions). A canonical decomposition of these groups is developed as an extension of an euclidean formalism and illustrated by a 5D example. Potential applications include in particular, moving reference frames and machine learning.