Prove that the Gaussian Integers are an integral domain

So the problem with your proof is that you assume that $ad-bc\ne 0$ and $ad+bc\ne 0$. This is not supported by your argument at all. However, if you note that $\Bbb Z[i]\subseteq\Bbb C$ you can use polar coordinates, write $z=r_1e^{i\theta_1}, w=r_2e^{i\theta_2}$ with $r_1, r_2>0$. Then their product is

$$zw= r_1r_2e^{i(\theta_1+\theta_2)}$$

which has absolute value $r_1r_2$. Since $r_1,r_2$ are real numbers which are not-zero, they multiple to a non-zero value. But if $|zw|\ne 0$ clearly $zw\ne 0$.