nilpotent elements in $\Bbb Z_n[i]$

Question

If $a^2+b^2$ is a nilpotent element of $\Bbb Z_n$ then $(a+bi)(a-bi)=a^2+b^2$ is also a nilpotent in $\Bbb Z_n[i]$. Does this imply that $a+bi$ or $a-bi$ is a nilpotent element in $\Bbb Z_n[i]$? Any idea would be helpful.