Is it not, in fact, intrinsically 2-dimensional? The enclosing contour that defines the prominence is a 2d entity. I don't know anything about persistent homology.

Maybe by 0-dimensional you mean that the nodes of the "contour tree" from which prominence can be calculated are labelled with scalars, not vectors. But isosurfaces of a scalar function in an arbitrarily high dimension would still be labelled with scalars. I guess that implies that persistent homology can be used with vector-valued data points to assess the topology of a vector field?

Another CS thing that's somewhat similar in flavour is scale space analysis.