It is challenging to identify and understand the qualitatively different regimes of the far-from-equilibrium dynamics of interacting quantum systems. In contrast to quantum or thermodynamic phase transitions, in quench dynamics it is often impossible to define an order parameter due to the lack of symmetry breaking, the evasion of thermalization, or the onset of chaos. Here we show that the time average of the Krylov complexity $C_K$, for a given Hamiltonian and initial state, can exhibit sharp, nonanalytic transitions that separate different regimes of dynamics as the coupling strength $g$ is tuned. We map out the complexity phase diagram for the Dicke model of light-matter interaction. At the transition, the slope of $C_K$ changes suddenly to coincide with a jump in the Krylov entropy. We elucidate the nature of the transition in Krylov space, where a particle is confined by a roughly linear potential but hops as if it lived in a Rindler reference frame, i.e., in the presence of synthetic gravity. The transition is then identified as the point where the oscillating wave packet gets destroyed by the extreme stretching, analogous to gravitational spaghettification, due to the Rindler hopping. This parameter-free, unambiguous framework for characterizing quench dynamics brings fresh insights to many interacting systems including quantum spin models.