  We theoretically study spin-$1/2$ fermions confined to two spatial dimensions
and experiencing isotropic short-range attraction in the presence of both
spin-orbit coupling and Zeeman spin splitting - a prototypical system for
developing topological superfluidity in the many-body sector. Exact solutions
for two-particle bound states are found to have a triplet contribution that
dominates over the singlet part in an extended region of parameter space where
the combined Zeeman- and center-of-mass-motion-induced spin-splitting energy is
large. The triplet character of dimers is purest in the regime of weak $s$-wave
interaction strength. Center-of-mass momentum is one of the parameters
determining the existence of bound states, which we map out for both two- and
one-dimensional types of spin-orbit coupling. Distinctive features emerging in
the orbital part of the bound-state wave function, including but not limited to
its $p$-wave character, provide observable signatures of unconventional
pairing.
