  We compute the topological phase diagram of 2D tetragonal superconductors for
the only possible nodeless pairing channels compatible with that crystal
symmetry. Subject to a Zeeman field and spin-orbit coupling, we demonstrate
that these superconductors show surprising topological features: non-trivial
high Chern numbers, massive edge states, and zero-energy modes out of high
symmetry points, even though the edge states remain topologically protected.
Interestingly, one of these pairing symmetries, $d+id$, has been proposed to
describe materials such as water-intercalated sodium cobaltates, bilayer
silicene or highly doped monolayer graphene, which opens the way for further
applications of our results.
