  Low temperature scanning tunneling microscopy is used to image the doped
topological insulator Bi$_{2-x}$Fe$_x$Se$_3$. Interstitial Fe defects allow the
detection of quasiparticle interference (QPI), and the reconstruction of the
empty state band structure. Quantitative comparison between measured data and
density functional theory calculations reveals the unexpected coexistence of
quantum well states (QWS) with topological surface states (TSS) on the
atomically clean surface of Bi$_{2-x}$Fe$_x$Se$_3$. Spectroscopic measurements
quantify the breakdown of linear dispersion due to hexagonal warping.
Nonetheless, both QWS and TSS remain spin-polarized and protected from
backscattering to almost 1 eV above the Dirac point, suggesting their utility
for spin-based applications.
