4-dimensional conformal field theories (CFTs) with N=2 superconformal symmetry have moduli spaces of vacua with spontaneously broken scale invariance which whose geometry is tightly constrained. All possible such geometries whose Coulomb branches are homeomorphic to a plane are constructed, and the global flavor symmetries of the associated CFTs are deduced. There are ten such CFTs, one of which was not found before by previous methods. We also construct examples of more exotic 1-complex-dimensional Coulomb branches with more complicated complex structures, but do not classify them.