[ugwprsop] Gaussian composites

What does the pattern of Gaussian integer multiples of a given complex number look like?  This should be easy.

Potentially nice demo: hover over a lattice of integers in the complex plane, and the multiples of the lattice point under the mouse cursor light up.  It will probably make nice symmetric patterns.  This would be useful for understanding the Gaussian primes, which aren't multiples of anything.