  Impurities in superconductors and their induced bound states are important
both for engineering novel states such as Majorana zero-energy modes and for
probing bulk properties of the superconducting state. The high-temperature
cuprates offer a clear advantage in a much larger superconducting order
parameter, but the nodal energy spectrum of a pure $d$-wave superconductor only
allows virtual bound states. Fully gapped $d$-wave superconducting states have
however been proposed in several cuprate systems thanks to subdominant order
parameters producing $d+is$- or $d+id'$-wave superconducting states. Here we
study both magnetic and potential impurities in these fully gapped $d$-wave
superconductors. Using analytical T-matrix and complementary numerical
tight-binding lattice calculations, we show that magnetic and potential
impurities behave fundamentally different in $d+is$- and $d+id'$-wave
superconductors. In a $d+is$-wave superconductor, there are no bound states for
potential impurities, while a magnetic impurity produces one pair of bound
states, with a zero-energy level crossing at a finite scattering strength. On
the other hand, a $d+id'$-wave symmetry always give rise to two pairs of bound
states and only produce a reachable zero-energy level crossing if the normal
state has a strong particle-hole asymmetry.
