A (algebraic) number field is defined to be a finite-degree extension field of $\mathbb{Q}$.
Let $f$ be a monic, irreducible polynomial in $\mathbb{Z}[X]$ and $\alpha$ be an complex-but-not-rational root of $f$. It is known that $\mathbb{Q}[\alpha]$ is a number field with extension degree of $\deg f$.
My question is that, for any number field $K$, if it's true that $K$ is isomorphic to some $\mathbb{Q}[\alpha]$ as defined above.
For the question exactly as stated, the answer is no: $\mathbb{Q}$ is a finite-degree extension field of $\mathbb{Q}$, and thus an algebraic number field, but it cannot be written as a field extension $\mathbb{Q}[\alpha]$ where $\alpha$ is irrational.
The answer becomes yes if you remove the clause "complex-but-not-rational". As noted in the comments, this is the content of the primitive element theorem when given the fact that $\mathbb{Q}$ is perfect, a consequence of being characteristic zero.