  We improve 1987 estimates of Patterson for sums of quartic Gauss sums over
primes. Our Type-I and Type-II estimates feature new ideas, including use of
the quadratic large sieve over $\mathbb{Q}(i)$, and Suzuki's evaluation of the
Fourier-Whittaker coefficients of quartic theta functions at squares. We also
conjecture asymptotics for certain moments of quartic Gauss sums over primes.
