  We model the newly predicted high-$T_c$ superconducting candidates
constructed by corner-shared trigonal bipyramidal complexes with an effective
three-orbital tight-banding Hamiltonian and investigate the pairing symmetry of
their superconducting states driven by electron-electron interactions. Our
combined weak and strong coupling based calculations consistently identify the
chiral $d+id$ superconductivity as the leading pairing symmetry in a wide
doping range with realistic interaction parameters. This pairing state has
nontrivial topological Chern-number and can host gapless chiral edge modes, and
the vortex cores under magnetic field can carry Majorana zero modes.
