It seems to me that there can be almost quaternionic structure on a manifold of two real dimensions, i.e., that generated by ($I,i\sigma_a$) where $I$ is the 2$\times$2 identity matrix and $\sigma_a$ are the Pauli matrices.
The three complex structures $J_a=i\sigma_a$ give rise to three Kaehler forms using $$ \Omega_a={(J_a)_{\mu}}^{\nu}g_{\nu\rho}dx^{\nu}\wedge dx^{\rho} $$ Why is it then true that we cannot have hyperkahler structure in two real dimensions?
Let $V$ be a finite-dimensional real vector space and let $I$, $J$, $K$ be linear maps $V \to V$ such that $I^2 = J^2 = K^2 = IJK = -\operatorname{id}_V$. We can view $V$ as a $\mathbb{H}$-module by defining the map $\mathbb{H}\times V \to V$ by
$$(a + bi + cj + dk, v) \mapsto av + bIv + cJv + dKv.$$
As modules over skew-fields are free (see this question), there is an isomorphism $V \cong \mathbb{H}^n$ for some $n$ and hence $\dim_{\mathbb{R}}V = 4n$.
If $M$ is a hyperKähler manifold, then the above argument shows that $T_pM$ has dimension $4n$ for some $n$, and therefore $M$ has dimension $4n$. In particular, $M$ cannot have dimension $2$.