Topological singularities are ubiquitous in many areas of physics. Polarization singularities are locations at which an aspect of the polarization ellipse of light becomes undetermined or degenerate. At C points, the orientation of the ellipse becomes degenerate and light’s electric field vector describes a perfect circle in time. In 2D slices of 3D random fields, the distribution in space of the C points is reminiscent of that of interacting particles. With near-field experiments, we show that when light becomes truly 2D, this has severe consequences for the distribution of C points in space. The most notable change is that the probability of finding two C points with the same topological index at a vanishing distance is enhanced in a 2D field. This case is an unusual finding for any system that exhibits topological singularities, as same-index repulsion is typically observed. All of our experimental findings are supported with theory, and excellent agreement is found between theory and experiment.