Nematic superconductivity with spontaneously broken rotation symmetry has recently been reported in doped topological insulators, M x Bi 2 Se 3 (M = Cu, Sr, Nb). Here we show that the electromagnetic (EM) response of these compounds provides a spectroscopy for bosonic excitations that reflect the pairing channel and the broken symmetries of the ground state. Using quasiclassical Keldysh theory, we find two characteristic bosonic modes in nematic superconductors: the nematicity mode and the chiral Higgs mode. The former corresponds to the vibrations of the nematic order parameter associated with broken crystal symmetry, while the latter represents the excitation of chiral Cooper pairs. The chiral Higgs mode softens at a critical doping, signaling a dynamical instability of the nematic state towards a new chiral ground state with broken time reversal and mirror symmetry. Evolution of the bosonic spectrum is directly captured by EM power absorption spectra. We also discuss contributions to the bosonic spectrum from subdominant pairing channels to the EM response.