The main question of the PI has been beautifully answered by Moret-Bailly, but not the secondary question arisen from its expectation: "I would expect that it should be possible to write down a power series in two variables f(x,y) so that ℂ[[x,y]]/f(x,y) is non-algebraic." though we got quite close in comments.
So for the record: this is not possible. Indeed, such a singularity would be analytic by Angelo's answer to [this MO question][1] (that is by a result of Michael Artin akin to the one I mentioned) and then algebraic by Ulrich's comment (that is by Corollary 7.7.3 of the book by Casas-Alvero "Singularities of plane curves", London Mathematical Society Lecture Note Series, 278).
This is of course consistent with the fact that the example quoted by Moret-Bailly is in three variables.