The integral closure of $\mathbb Z$ in $\mathbb Q$ the set of elements in $\mathbb Q$ which satisfy a monic polynomial over $\mathbb Z$. This is $\mathbb Z$.
The integral closure of $\mathbb Z$ in $\mathbb Q[i]$ the set of elements in $\mathbb Q[i]$ which satisfy a monic polynomial over $\mathbb Z.$ This is $\mathbb Z[i]$.
The integral closure of $\mathbb Z$ in $\mathbb Q[\sqrt 3]$ the set of elements in $\mathbb Q[\sqrt 3]$ which satisfy a monic polynomial over $\mathbb Z$. This is $\mathbb Z[\sqrt 3]$.
Is the integral closure of $\mathbb Z[i]$ in $\mathbb Q[i]$ the set of elements in $\mathbb Q[i]$ which satisfy a monic polynomial over $\mathbb Z[i]$? In other words, is it the set of elements in $\mathbb Q[i]$ which satisfy a monic polynomial in $\mathbb Z[i][x]$?
What is the integral closure of $\mathbb Z[i]$ in $\mathbb Q[i]$? In other words, what elements in $\mathbb Q[i]$ satisfy a monic polynomial whose coefficients are in $\mathbb Z[i]$?