A quaternionic matrix $A$ gives rise to a function $\mathbb{H}^n \to \mathbb{H}^n$ given by $x \mapsto A \cdot x$. This is real linear, but not complex- or quaternionic-linear (in general) if we consider $\mathbb{H}^n$ as a left $\mathbb{C}$ or $\mathbb{H}$ module, but is pretty good if we use right actions.

A right eigvenvalue of $A$ is a quaternion $q$ such that $A\cdot x = x \cdot q$ for some $x\in \mathbb{H}^n$; a left eigenvalue is quaterion $q$ such that $A \cdot x = q\cdot x$ for some $x\in \mathbb{H}^n$.

The algebra of right eigenvalues is pretty good, but the algebra of left eigenvalues is quite interesting. For example, it is not hard to see that there are matrices $A$ with infinitely many left eigenvalues, even for $2$-by-$2$ matrices.

Now let's assume that $A\in Sp(n)$, so that the left eigenvalues are all contained in $S^3\subseteq \mathbb{H}$. What sort of geometric properties must the set $L(A)$ of left eigenvalues have?

EDIT: An example is $$ \begin{pmatrix} 0 & 1 \cr -1 & 0 \end{pmatrix} \cdot \begin{pmatrix} 1 \cr q \end{pmatrix} = q \cdot {\begin{pmatrix} 1 \cr q \end{pmatrix}}$$ for any $q\in S^3 \subseteq \mathbb{H}$ with zero real part, since then $q^2 = -1$.

EDIT 2: Examples like this show that for some symplectic matrices, the set of left eigenvalues is a union of copies of $S^2$.