Emil's idea about quantifier elimination is a good one.

The paper "Model Theory of valued fields" by Raf Cluckers cites the thesis "Quelques propriétés des corps valués", which I wasn't able to find online, for the claim that the field $\mathbb C((t))$ admits quantifier elimination for the language of Macintyre which consists of the language of field theory, a symbol for elements of the valuation ring $\mathbb C[[t]]$, and a symbol for $n$th powers. Thus any set definable in your sense is definable, without quantifiers, in this language.

It is clear that any set definable in this language has only finitely many isolated points and therefore cannot be $\mathbb C$ or $\mathbb C[x]$.

Emil's idea about quantifier elimination is a good one.

The paper "Model Theory of valued fields" by Raf Cluckers cites the thesis "Quelques propriétés des corps valués" by F. Delon, which I wasn't able to find online, for the claim that the field $\mathbb C((t))$ admits quantifier elimination for the language of Macintyre which consists of the language of field theory, a symbol for elements of the valuation ring $\mathbb C[[t]]$, and a symbol for $n$th powers. Thus any set definable in your sense is definable, without quantifiers, in this language.

It is clear that any set definable in this language has only finitely many isolated points and therefore cannot be $\mathbb C$ or $\mathbb C[x]$.