A chain of Josephson junctions represents one of the simplest many-body models undergoing a superconductor-insulator quantum phase transition(1,2). Apart from zero resistance, the superconducting state is necessarily accompanied by a sound-like mode due to collective oscillations of the phase of the complex-valued order parameter(3,4). Little is known about the fate of this mode on entering the insulating state, where the order parameter s amplitude remains non-zero, but the phase ordering is melted by quantum fluctuations(5). Here, we show that the phase mode survives far into the insulating regime, such that megaohm-resistance chains can carry gigahertz-frequency alternating currents as nearly ideal superconductors. The insulator reveals itself through interaction-induced broadening and random frequency shifts of collective mode resonances. Our spectroscopic experiment puts forward the problem of quantum electrodynamics of a Bose glass for both theory and experiments(6-8). By pushing the chain parameters deeper into the insulating state, we achieved a wave impedance of the phase mode exceeding the predicted critical value by an order of magnitude(9-14). The effective fine structure constant of such a one-dimensional electromagnetic vacuum exceeds unity, promising transformative applications to quantum science and technology.