We know $\mathbb{Z}[x]/(x^2+1)\cong \mathbb{Z}[i]$. From this, I guess a similar thing happens, $$\mathbb{Q}[x,y]/(x^2+y^2)\cong \mathbb{Q}[y,yi]$$ by a map $F:\mathbb{Q}[x,y]\rightarrow \mathbb{Q}[y,yi]$ mapping $y\rightarrow y, x\rightarrow yi$. I think I could prove the kernel of $F$ is $(x^2+y^2)$ but I am not sure. Is the argument true?
I got this question from this post, What is the field of fractions of $\mathbb{Q}[x,y]/(x^2+y^2)$?. And if the above mapping is true, we get $\mathrm{Frac}(\mathbb{Q}[x,y]/(x^2+y^2))=\mathrm{Frac}(\mathbb{Q}[y,yi])=\mathbb{Q}(y,yi)=\mathbb{Q}(y,i)$
The standard monomials of ${\Bbb Q}[x,y]/\langle x^2+y^2\rangle$ are $1,y,y^2,y^3,\ldots$ and $x$ with $x^2=-y^2$ and so $x=\pm iy$. This gives certainly an isomorphism of vector spaces with ${\Bbb Q}[y,iy]$. That it is also an isomorphism of rings requires a proof: $y\mapsto y$ and $x\mapsto iy$.
More insight into standard monomials is provided by Cox, Little, O'Shea ''Using Algebraic Geometry'' and the more elementary text of Cox, Little, O'Shea ''Ideals, Varieties, and Algorithms''.