Let $k$ be a field and $f\in k[a,b]$ an irreducible two-variable polynomial, $B := k[a,b]/(f)$ and $C$ the integral closure of $B$ in its fraction field.
I call $f$ good if it is irreducible in the ring $k[[a,b]]$ of formal power series; equivalently (Nagata, Local Rings, p. 122, Ex. 1), if $C$ has exactly one prime ideal lying above $\mathfrak m = (a,b)B$ (hence, $f$ being good just means that the curve $B$ is analytically irreducible at the origin).
I'm looking for examples of "good" polynomials $f$ such that the residue field $L$ of $C$ is a proper extension of $k$.
I can show (at least if $k$ is infinite) that $[L:k] \cdot r = \mu(f)$, where $\mu(f)$ is the degree of the lowest-degree summand of $f$, and $r$ is the ramification index of $\mathfrak m$ in $C$. Hence, it is clear that $\mu(f)$ must be large enough if one wants interesting examples.
The only "generic" class of examples I could come up with is the one where $f$ is homogeneous: Then $f$ being irreducible implies $f$ is good, and $[L:k] = \deg f$.