  We define a `Higgs field' for a four-dimensional spin$^c$-manifold to be a
smooth section of its positive half-spinor bundle, transverse to the zero
section, and defined only up to a positive functional factor. This is intended
to be a generalization of almost complex structures on real four-manifolds,
each of which may in fact be treated as a Higgs field without zeros for a
specific spin$^c$-structure. The notions of totally real or pseudoholomorphic
immersions of real surfaces in an almost complex manifold of real dimension
four have straighforward generalizations to the case of a spin$^c$-manifold
with a Higgs field.
  Our results consist, first, in showing that totally real immersions of closed
oriented surfaces in four-dimensional spin$^c$-manifolds with Higgs fields
have, basically, the same properties as in the almost-complex case, and,
secondly, in providing a description of all pseudoholomorphic immersions of
such surfaces in the four-sphere endowed with a "standard" Higgs field.
