The counterexamples of normal extensions over $\mathbb{Q}$ that I could find are basically all in the form of $E = \mathbb{Q}(\sqrt[n]{\alpha})$ (with $x^n - \alpha$ being its minimal polynomial; and the minimal polynomial has complex roots, so doesn't split in $\mathbb{Q}(\sqrt[n]{\alpha})$).
I wonder if there is example such that the extension $E/\mathbb{Q}$ that is both non normal, and also contains complex numbers.
Any hint of how to find such extension would be appreciated.