*This is really a comment, but it's far too long to go into a comment window.*

Your questions need to be made more precise.

First, if $(M,J)$ is complex and $g$ is a metric on $M$ that is $J$-Hermitian, there are (at least) two possible connections that you can associate with $(M,J,g)$ and the different connections can have different holonomies. There is the Levi-Civita connection $\nabla^g$ of $g$ (which may not have $\nabla^g J = 0$), and, second, there is a canonical connection $\nabla^{J,g}$ for which $J$ and $g$ are parallel but has torsion of type (1,1). These are equal iff $(M,J,g)$ is Kähler.

For which of these are you asking about the holonomy? (There is actually a whole line of connections joining the two when $(M,J,g)$ is not Kähler, as the space of connections on the tangent bundle is an affine space, so you could be asking about some connection that is a combination of these. Actually, there are even more than that associated to the triple $(M,J,g)$, but let's not go there.)

Second, I don't know what you mean, in the hypercomplex case, by the statement that "Each complex structure will admit three Hermitian metrics." Where are these metrics coming from? They can't naturally arise from any construction using the three anti-commuting complex structures alone. For example, consider $M= \mathbb{H}^n$ with its natural hypercomplex structure given by multiplication on the *right* by $i$, $j$, and $k$. The group $\mathrm{GL}(n,\mathbb{H})$ acts on the left (by matrix multiplication preserving the hypercomplex structure), but it does not preserve any metric.