Almost Kähler Walker 4-manifolds

It is shown that any proper almost Hermitian structure on a Walker 4-manifold is isotropic Kähler. Moreover, a local description of proper almost Kähler structures that are self-dual, ∗∗-Einstein or Einstein is given and it is proved that any proper strictly almost Kähler Einstein structure is self-dual, Ricci flat and ∗∗-Ricci flat. This is used to supply examples of flat indefinite non-Kähler almost Kähler structures.