  We prove a structure theorem for multiplicative functions on the Gaussian
integers, showing that every bounded multiplicative function on the Gaussian
integers can be decomposed into a term which is approximately periodic and
another which has a small U^{3}-Gowers uniformity norm. We apply this to prove
partition regularity results over the Gaussian integers for certain equations
involving quadratic forms in three variables. For example, we show that for any
finite coloring of the Gaussian integers, there exist distinct nonzero elements
x and y of the same color such that x^{2}-y^{2}=n^{2} for some Gaussian integer
n. The analog of this statement over Z remains open.
