TITLE: Wanted: example of a non-algebraic singularity
QUESTION [31 upvotes]: Given a finitely generated $\def\CC{\mathbb C}\CC$-algebra $R$ and a $\CC$-point (maximal ideal) $p\in Spec(R)$, I define the singularity type of $p\in Spec(R)$ to be the isomorphism class of the completed local ring $\hat R_p$, as a $\CC$-algebra.
Do there exist non-algebraic singularity types? That is, does there exist a complete local ring with residue field $\CC$ which is formally finitely generated (i.e. has a surjection from some $\CC[[x_1,\dots, x_n]]$), but is not the complete local ring of a finitely generated $\CC$-algebra at a maximal ideal?
Googling for "non-algebraic singularity" suggests that the answer is yes, but I can't find a specific example. I would expect that it should be possible to write down a power series in two variables $f(x,y)$ so that $\CC[[x,y]]/f(x,y)$ is non-algebraic.

What is a specific formally finitely generated non-algebraic singularity?

REPLY [15 votes]: The main question of the PI has been beautifully answered by Moret-Bailly, but not the
secondary question arisen from his 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 a result of Michael Artin ( "On the solutions of analytic equations", Invent. Math. 5 1968, 277–291, cf. Angelo's answer to Analytic vs. formal vs. étale singularities) 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.