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Proving that two fields are isomorphic

I have to show that the rational number field and the complex numbers are isomorphic. This is my attempt:
By the definition of isomorphic, we must show that $\forall x,y \in \mathbb{Q}$, $f(x)=f(y)$ where $f$ is the isomorphism. Then we should show that the only two fields that have the property are $\mathbb{Q}$ and the complex numbers.
I’m really lost here, since I don’t know how to show this. Any hint or help is appreciated,
Thanks!

Let $f$ be the isomorphism. Let $x,y \in \mathbb{Q}$. It is easy to see that $f(x)=f(y)$ if and only if $f(x) \in \mathbb{C}$ and $f(y) \in \mathbb{C}$.

In the next step, show that there are only two fields which have this property.

Hint: If there are more than one field with this property, there will be two fields with the same cardinality.

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