I will discuss my recent work constructing a non-conical singular Hermitian Yang-Mills connection on a homogeneous reflexive sheaf over \mathbb{C}^3, which is supposed to model the generic situation of bubbling phenomenon when the Fueter section has a zero. This example in particular shows that the uniqueness part of the Hitchin-Kobayashi correspondence does not extend naively to noncompact manifolds. A variant of this construction gives a sequence of HYM connections on the unit ball in \mathbb{C}^3 with uniformly bounded L^2 curvature, but the number of codimension 6 singularities tends to infinity along the sequence. This illustrates the substantial difficulty of the compactification problem in higher dimensional gauge theory.