We use functions of a bicomplex variable to unify the existing constructions of harmonic morphisms from a 3-dimensional Euclidean space to a Riemannian or Lorentzian surface. This is done by using the notion of complex harmonic morphism between complex-Riemannian manifolds and showing how these are given by bicomplex functions when the codomain is one-bicomplex dimensional. By taking real slices, we recover well-known compactifications for the three possible real cases. On the way, we discuss some interesting conformal compactifications of complex-Riemannian by interpreting them as bicomplex manifolds. This led to interesting examples of globally defined harmonic morphisms other than orthogonal projection and harmonic morphisms all of whose fibers are degenerate and it was shown that such degenerate harmonic morphisms correspond to real-valued null solutions of the wave equations. Many notions and results for harmonic morphisms between semi-Rimannian manifolds complexify immediately to complex –harmonic morphisms between complex-Rimannian manifols and giving a way of constructing complex-harmonic morphisms into a One-dimensional bicomplex manifolds. These aremany complex harmonic morphisms from open subsets of C^3 to C^2=B which are not obtained by extending a real harmonic morphisms. Harmonic morphisms into Rimannian or Lorentzian syrfaces are particulary nice as they are conformally invariant in the sense that only the conformal equivalence and C^2 solutio q=?(z) to this equation are complex harmonic morphism from open subsets of C^3 to N, and all such harmonic morphisms which are submersive are given this way, locally. In general, this equationdefines a congruence of lines and planes. Indeed, for each q? N, if CN(G) ?-1, this equationdefines a complex line, whereas if CN(G)=-1, either has no solution or defines a plane, we shall call these lines and planes the fiber of the congruence this equation as they form the fibers of any smooth harmonic morphism q=?(z) which that equation. How ever, starting with arbitrary data G and H the fiber of the congruence that equatio may intersect or have envelope points where they become infinitesimally close. We shall consider the behavior of this congruence when the fibers are degenerate or have direction not represented by a finite value of G.