We construct a time-dependent double well potential as an exact spectral equivalent to the explicitly time-dependent negative quartic oscillator with a time-dependent mass term. Defining the unstable anharmonic oscillator Hamiltonian on a contour in the lower-half complex plane, the resulting time-dependent non-Hermitian Hamiltonian is first mapped by an exact solution of the time-dependent Dyson equation to a time-dependent Hermitian Hamiltonian defined on the real axis. When unitary transformed, scaled and Fourier transformed we obtain a time-dependent double well potential bounded from below. All transformations are carried out non-perturbatively so that all Hamiltonians in this process are spectrally exactly equivalent in the sense that they have identical instantaneous energy eigenvalue spectra. •Time-dependent spectrally equivalent potentials are constructed.•Double wells are equivalent to time-dependent negative quartic oscillators.•We provide new analytical solutions to the time-dependent Dyson equation.