More directly.
1)
2)
multiply 1) by d on both sides
Apply 2
Case1) b=0 then your first one a+bi is just an integer which is definitely not a zero divisor, I think you can check that easily enough
Case2) c^2+d^2=0 but this means both c and d must be 0 because integers squared are nonnegative. But that is exactly the condition that the second gaussian integer was 0.