Spinors as automorphisms of the tangent bundle

We show that, on a $4$-manifold $M$ endowed with a $\operatorname{spin}^{\mathbb{C} }$-structure induced by an almost-complex structure, a self-dual (positive) spinor field $\phi\in\Gamma(W^+)$ is the same as a bundle morphism $\phi:T_M\to T_M$ acting on the fiber by self-dual conformal transformations, such that the Clifford multiplication is just the evaluation of $\phi$ on tangent vectors, and that the squaring map $\sigma:\mathcal{W}^+\to\Lambda^+$ acts by pulling-back the fundamental form of the almost-complex structure. We use this to detect Kähler and symplectic structures.