  In the framework of quaternionic Clifford analysis in Euclidean space
$\mathbb{R}^{4p}$, which constitutes a refinement of Euclidean and Hermitian
Clifford analysis, the Fischer decomposition of the space of complex valued
polynomials is obtained in terms of spaces of so--called (adjoint) symplectic
spherical harmonics, which are irreducible modules for the symplectic group
Sp$(p)$. Its Howe dual partner is determined to be $\mathfrak{sl}(2,\mathbb{C})
\oplus \mathfrak{sl}(2,\mathbb{C}) = \mathfrak{so}(4,\mathbb{C})$.
