  We show the existence of a modified Cliff(1,1) structure compatible with an
Osserman 0-model of signature (2,2). We then apply this algebraic result to
certain classes of pseudo-Riemannian manifolds of signature (2,2). We obtain a
new characterization of the Weyl curvature tensor of an (anti-)self-dual
manifold and we prove some new results regarding (Jordan) Osserman manifolds.
