give a definition of a quaternionic derivative, the so-called Cullen derivative [@Gra1; @Gra2], which is more useful for our purposes and we adopt it here.

The novelty of the present paper can be summarized as follows: using the Cullen derivative we define the quaternionic counterparts of the Hermite polynomials $H_n(z)$ and $H_{n,m}(z,\overline{z})$, as vectors in quaternionic Hilbert spaces by replacing $z$ by a quaternion ${\mathbf{q}}$. The two index quaternionic Hermite polynomials $H_{n,m}({\mathbf{q}},\overline{{\mathbf{q}}})$ will span a subspace $\mathfrak H_{\mathbf{q}}$ of a quaternionic $L^2$-space. By fixing $n=0$ or $m=0$ in $H_{n,m}({\mathbf{q}},\overline{{\mathbf{q}}})$ we shall obtain the regular and anti-regular subspaces of $\mathfrak H_{\mathbf{q}}$ (as the counterparts of holomorphic and anti-holomorphic subspaces). We shall also prove that the single indexed quaternionic Hermite polynomials $H_n({\mathbf{q}})$ and $H_n(\overline{{\mathbf{q}}})$ serve as bases for certain regular and anti-regular subspaces respectively. Apart from these, by defining kernels with $H_n({\mathbf{q}})$ and $H_{n,m}({\mathbf{q}},\overline{{\mathbf{q}}})$ we will obtain CS resembling (\[ma-cs\]) over quaternionic Hilbert spaces and realize the regular subspace as a reproducing kernel Hilbert space. The functions $e^{-|{\mathbf{q}}|^2/2}H_{n,m}({\mathbf{q}},\overline{{\mathbf{q}}})$ are shown to be eigenfunctions of an operator $\mathfrak{L}_H$ with infinite degeneracy as in the Landau problem [@Thi2] (see also end of Section \[sec-quherm-polyn\]). However, we are unable at this point to give a 