Here are some small facts about Quaternionic Geometry. First let us leave metrics aside, for the moment. There are 2 analogues of complex manifolds in the quaternionic world, namely triholomorphic manifolds and quaternionic manifolds. They are very different. Triholomorphic manifolds admit a 2-sphere of complex structures. When I write complex structure, I mean integrable almost complex structure. On the other hand, quaternionic manifolds are manifolds admitting an almost quaternionic structure satisfying an integrability condition. They may not admit a 2-sphere of almost complex structures globally, though they do locally. But these almost complex structures need not be integrable.

As an example of a triholomorphic manifold, think of $\mathbb{H}^n$ for instance. An example of a quaternionic manifold would be $\mathbb{H}P^n$, which does not even admit a globally defined almost complex structure.

The metric analogues of these 2 classes of manifolds, would be hyperkähler manifolds and quaternion-kähler manifolds. Their definitions can be found in many places (wikipedia for instance, or Besse's Einstein manifolds, or books or lecture notes by say, Salamon for instance).

I hope this will point you in the right direction.