Pseudo-Riemannian spinc manifolds were introduced by Ikemakhen in . In the present work we consider pseudoRiemannian 4-manifolds with neutral signature whose structure groups are SO+(2; 2). We prove that such manifolds have pseudoRiemannian spinc structure. We construct spinor bundle S and half-spinor bundles S+ and S-on these manifolds. For therst Seiberg-Witten equation we define Dirac operator on these bundles. Due to the neutral metric self-duality of a 2-form is meaningful and it enables us to write down second Seiberg-Witten equation. Lastly we write down the explicit forms of these equations on 4-dimensional at space.