Recall that if G is a group, k a field, and V_k an irreducible representation of G over k then End_G(V_k) is a division algebra D over k. For example, if $V_{\mathbb{R}}$ is a real representation then the endomorphism ring is $\mathbb{R}$, $\mathbb{C}$, or $\mathbb{H}$. Such representations are typically called real, complex, or quaternionic.

This usage is complimentary to your definition. Suppose that $V_{\mathbb{R}}$ is a quaternionic representation, then $\mathbb{H}$ acts on $V_{\mathbb{R}}$ on the right. Since $\mathbb{H}$ is a division ring, it follows that $V_{\mathbb{R}} \cong \mathbb{H}^k$. Hence a real representation which is quaternionic is canonically a quaternionic representation in your sense.

Now Frobenius-Schur indicator theory explains how to determine which representations over $\mathbb{C}$ "come from" which kind of real representations. Explicitly, if $V_{\mathbb{C}}$ is not selfdual then it plus its dual is the complexification of a representation over $\mathbb{R}$ with $D=\mathbb{C}$; if $V_{\mathbb{C}}$ orthogonal (i.e. has an invariant bilinear form) then it is the complexification of a representation over $\mathbb{R}$ with $D=\mathbb{R}$; finally, if $V_{\mathbb{C}}$ is symplectic then it plus itself is the complexification of a representation over $\mathbb{R}$ with $D=\mathbb{H}$.

In summary, given any symplectic representation $V_{\mathbb{C}}$ the representation $V_{\mathbb{C}} \oplus V_{\mathbb{C}}$ is the complexification of a representation $W_\mathbb{R}$ which is itself a module over $\mathbb{H}$. In particular, there is an "interesting" representation over $\mathbb{H}$ whose quaternionic dimension is half the complex dimension of $V_{\mathbb{C}}$. In particular, the Lie group $Sp(2n)$ should have an interesting n-dimensional representation over the quaternions.

Recall that if G is a group, k a field, and V_k an irreducible representation of G over k then End_G(V_k) is a division algebra D over k. For example, if $V_{\mathbb{R}}$ is a real representation then the endomorphism ring is $\mathbb{R}$, $\mathbb{C}$, or $\mathbb{H}$. Such representations are typically called real, complex, or quaternionic.

This usage is complimentary to your definition. Suppose that $V_{\mathbb{R}}$ is a quaternionic representation, then $\mathbb{H}$ acts on $V_{\mathbb{R}}$ on the right. Since $\mathbb{H}$ is a division ring, it follows that $V_{\mathbb{R}} \cong \mathbb{H}^k$. Hence a real representation which is quaternionic is canonically a quaternionic representation in your sense.

Now Frobenius-Schur indicator theory explains how to determine which representations over $\mathbb{C}$ "come from" which kind of real representations. Explicitly, if $V_{\mathbb{C}}$ is not selfdual then it plus its dual is the complexification of a representation over $\mathbb{R}$ with $D=\mathbb{C}$; if $V_{\mathbb{C}}$ orthogonal (i.e. has an invariant bilinear form) then it is the complexification of a representation over $\mathbb{R}$ with $D=\mathbb{R}$; finally, if $V_{\mathbb{C}}$ is symplectic then it plus itself is the complexification of a representation over $\mathbb{R}$ with $D=\mathbb{H}$.

In summary, given any symplectic representation $V_{\mathbb{C}}$ the representation $V_{\mathbb{C}} \oplus V_{\mathbb{C}}$ is the complexification of a representation $W_\mathbb{R}$ which is itself a module over $\mathbb{H}$. In particular, there is an "interesting" representation over $\mathbb{H}$ whose quaternionic dimension is half the complex dimension of $V_{\mathbb{C}}$. Your example fits into this scheme, where you start with the 2-dimensional irrep of $\mathbf{Q}$. Another great example involves the symplectic Lie group which can also be realized as a unitary group over the quaternions (see wikipedia).