  Recent studies have brought $\alpha$-RuCl$_3$ to the forefront of
experimental searches for materials realizing Kitaev spin-liquid physics. This
material exhibits strongly anisotropic exchange interactions afforded by the
spin-orbit coupling of the 4d Ru centers. We investigate the dynamical response
at finite temperature and magnetic field for a realistic model of the magnetic
interactions in $\alpha$-RuCl$_3$. These regimes are thought to host
unconventional paramagnetic states that emerge from the suppression of magnetic
order. Using exact diagonalization calculations of the quantum model
complemented by semi-classical analysis, we find a very rich evolution of the
spin dynamics as the applied field suppresses the zigzag order and stabilizes a
quantum paramagnetic state that is adiabatically connected to the fully
polarized state at high fields. At finite temperature, we observe large
redistributions of spectral weight that can be attributed to the anisotropic
frustration of the model. These results are compared to recent experiments, and
provide a roadmap for further studies of these regimes.
