The space of affine connections of an almost Hermitian manifold

We consider affine connections determined by an almost Hermitian structure of a smooth manifold. We prove that the affine space of connections considered has dimension $12$ if and only if the Lie form of the almost Hermitian structure is nonzero. We find connections that define post-Riemannian geometries and almost Hermitian connections in the class $W_4$. We examine a conformal transformation of an almost Hermitian structure and an affine mapping of connections generated by this transformation and find a connection invariant under this mapping.