  Topological superconductivity (TSC) is a promising platform to host Majorana
zero mode (MZM) for topological quantum computing. Recently, the noble metal
alloy AuSn$_{4}$ has been identified as an intrinsic surface TSC. However, the
atomic visualization of its nontrivial surface states and MZM remains elusive.
Here, we report the direct observation of unconventional surface states and
vortex zero mode at the gold (Au) terminated surfaces of AuSn$_{4}$, by
ultra-low scanning tunneling microscope/spectroscopy. Distinct from the trivial
metallic bulk states at tin (Sn) surfaces, the Au terminated surface exhibits
pronounced surface states near Fermi level. Our density functional theory
calculations indicate that these states arise from unconventional Rashba bands,
where two Fermi circles from different bands share identical helical spin
textures, chiralities, and group velocities in the same direction. Furthermore,
we find that although the superconducting gap, critical temperature,
anisotropic in-plane critical field are almost identical on Au and Sn
terminated surfaces, the in-gap bound states inside Abrikosov vortex cores show
significant differences. The vortex on Sn terminated surfaces exhibits a
conventional Caroli-de Gennes-Matricon bound state while the Au surface shows a
sharp zero-energy core state with a long non-splitting distance, resembling an
MZM in a non-quantum-limit condition. This distinction may result from the
dominant contribution of unconventional Rashba bands near Fermi energy from Au
terminated surface. Our results provide a new platform for studying
unconventional Rashba band and MZM in superconductors.
