This is very much like Factoring polynomials to factors involving complex coefficients except that I'm concerned about bivariate polynomials, not univariate polynomials. Take for example the polynomial
$$3 - 6 x^2 + 3 x^4 - 10 y^2 + 6 x^2 y^2 + 3 y^4$$
I'd like to factor that over the complex numbers. If I know where this came from, I can do Factor[poly, Extension -> {Sqrt[3]}]
to find that the above equals
$$\tfrac13 (3 - 3 x^2 + 2 \sqrt3 y - 3 y^2) (3 - 3 x^2 - 2 \sqrt3 y - 3 y^2)$$
but to find this I have to know, a priori, that $\sqrt3$ might be a useful extension.
Even more important, that I haven't forgotten any additional extensions which would have allowed me to factor this even more. (The latter is easy in the case above, but hard if the degrees get higher.)
Edit: I found that this question can be answered using IrreduciblePolynomialQ[poly, Extension → All].
Is there any way I can compute a factorization of a polynomial from $\mathbb Z[x,y]$ into factors from $\mathbb C[x,y]$ (or equivalently $\bar{\mathbb Q}[x,y]$ i.e. with algebraic coefficients), without the use of any knowledge except for the polynomial itself?