$\newcommand\Q{\mathbb{Q}} \newcommand\R{\mathbb{R}} \newcommand\C{\mathbb{C}}$
Consider the condition: $\alpha\in \R$ is an algebraic irrational real number and $\Q(\alpha)$ is not Galois (or normal) over $\Q$ and the splitting field, say $K$, of the minimal polynomial of $\alpha$ over $\Q$ has non-real elements (in this case it is a totally imaginary field).
Now, is it true that for all $\alpha$ satisfying above there is a $\beta\in K\backslash\R$ such that $\textrm{Im}(\beta)^2 \in \Q(\alpha)$?
I think the above can be answered by knowing a little bit more about quadratic field extensions (than I do). Any ideas or counterexamples?
Edit: My feeling is that we get a counterexample if we look at some splitting field of $x^n-q$ for some positive rational ($\neq 1$). Let's say the splitting field of $x^3-2$, then any non-real element $\beta$ of this field will not have the described property. This wouldn't work for an even degree $n$ divisible by $4$. Because you just take the purely imaginary $g^{n/4}$, where $g$ is a primitive $n$-th root of $2$, as your $\beta$.