This proposal is part of our research programme to explore the spin dynamics of strongly-frustrated quantum magnets. A canonical model is the spin-1/2 triangular antiferromagnet for which theoretical predictions for quantum dynamics effects leading to roton-like soft minima in the dispersion relation and extended scattering continua remain experimentally unexplored. In recent single-crystal measurements on LET we found that Ba3CoSb2O9 is a unique realization of such as system with i) strong renormalization of the spin-wave dispersion relation and zone-boundary soft modes ii) unexpectedly strong continuum scattering at intermediate to high energies. Here we propose to measure the magnetic field and temperature dependence of the dispersions and continuum scattering to compare with theories of quantum dynamics and distinguish between two-magnon and two-spinon continuum scattering.