Discriminant to Find Nature of Roots

Let $a,b,c$ be real numbers and $f(x)=ax^2+bx+c$ be a polynomial. Suppose that for any non-real complex number $x$, the value $f(x)$ is also non-real. (In other words: whenever $f(x)$ is real, so is $x$ itself.)

What are the possible values of $a$?