I have some general idea of going about this. I'm trying to come up with a suitable mapping. I've come up with $\phi\colon a + \langle x^2 + 1\rangle\longmapsto a$ where $a$ is an irreducible element in $\mathbb{Z}[i]$ but this mapping has some issues. For example take $5x + x^2 + 1$, which is $5x \pmod{x^2+1}$. $5x$ does not belong to $\mathbb{Z}[i]$. So my question is, is there a better mapping?
I know for a fact that if you add any integer call it $b$ to $\langle x^2 + 1\rangle$, you will get $b$ back which belongs to $\mathbb{Z}[i]$. I'm stuck when $b$ is not an integer.