  We analyse the $L^2$ Hilbert complexes naturally associated to a non-compact
complex manifold, namely the ones which originate from the Dolbeault and the
Aeppli-Bott-Chern complexes. In particular we define the $L^2$
Aeppli-Bott-Chern Hilbert complex and examine its main properties on general
Hermitian manifolds, on complete K\"ahler manifolds and on Galois coverings of
compact complex manifolds. The main results are achieved through the study of
self-adjoint extensions of various differential operators whose kernels, on
compact Hermitian manifolds, are isomorphic to either Aeppli or Bott-Chern
cohomology.
