  We prove that an isometric immersion of a timelike surface in
four-dimensional Minkowski space is equivalent to a normalized spinor field
which is a solution of a Dirac equation on the surface. Using the quaternions
and the complex numbers, we obtain a spinor representation formula that relates
the spinor field and the isometric immersion. Applying the representation
formula, we deduce a new spinor representation of a timelike surface in
three-dimensional De Sitter space; we give a formula for the Laplacian of the
Gauss map of a minimal timelike surface in four-dimensional Minkowski space in
terms of the curvatures of the surface; we obtain a local description of a flat
timelike surface with flat normal bundle and regular Gauss map in
four-dimensional Minkowski space, and we also give a conformal description of a
flat timelike surface in three-dimensional De Sitter space.
