We know that $\mathbb{C}$ is an algebraically closed topological field of characteristic zero with respect to the Euclidean topology, which is induced by the (complete) absolute value on $\mathbb{C}$. If one were to adjoin an extra point, say $\alpha$, to $\mathbb{C}$, then clearly $\overline{\mathbb{C}(\alpha)}$, the algebraic closure of $\mathbb{C}(\alpha)$, is an algebraically closed transcendental extension of $\mathbb{C}$.
Is it possible to extend the Euclidean topology to $\overline{\mathbb{C}(\alpha)}$ in such a way that addition, multiplication and inversion are continuous? (so that the extension is also a topological field). If so, is this topology induced by some norm?
Any pointers would be very helpful.