Let w be in Z(square root of -3) with w not equal to 0,1,-1. Prove w can be factored into irreducible elements of Z(square root of -3); that is, we can find irriducible elements p1,p2,...,pt with w=p1p2...pt
But I think this can be shown by induction on |w| (which is a positive integer). Simply by the argument, that it is either irreducible, or it can be written as a product of two elements with smaller absolute value.
Of course, you'll have to show that all units in this ring have absolute value 1 first, but this is easy.