In this article, our main goal is to extend the concept of dynamical sampling to quaternionic Hilbert spaces. The problem is to recover the initial state of an evolving function \(\mathfrak {f}\) from the system \(\mathcal {T}:=\{T^{n}\mathfrak {f}(m):n\in \Omega _m\text {, }m\in \mathcal {J}\}\) of its spatial sampling taken at different time levels n, where \(\mathcal {J}\) is an index set and \(\Omega _m=\{0\text {, }1\text {,}\dots \text {, }k_m\}\) . Initially the finite dimensional case for \(\mathbb {H}^d\) is studied and some characterizations for the system \(\mathcal {T}\) to form a frame for \(\mathbb {H}^d\) for diagonal and Hermitian matrices T are given. Further, these characterization for the infinite dimensional case where the operator is assumed to be diagonalizable are proved. Furthermore, we prove that for the case of a finite index set \(\mathcal {J}\) , the system \(\mathcal {T}\) never forms a minimal set and hence a basis for a right-quaternionic Hilbert space \(\mathcal {H}\) . Also, some specific conditions under which \(\mathcal {T}\) forms a frame for \(\mathcal {H}\) with finite set J are discussed. Finally, some characterizations for the system \(\mathcal {T}\) to form a frame for \(\mathcal {H}\) for general bounded operators in terms of strongly stable contractions are given.
