In my algebraic geometry class, the professor gave as an example that in general, two pairs of 4 points in $\mathbb{A}^1$ are not isomorphic as algebraic sets. He proved it for $\mathbb{C}$, where isomorphisms of algebraic sets are given by mobius transofrmations, and they preserve the cross-ratio. So for example, $V_1=\{1,2,3,4\}$ and $V_2=\{1,-1,i,-2i\}$ will not be isomorphic if I am not mistaken, since the cross ratio for the first will be real and for the second not (they do no lie on a line or a circle).
While this explanation is fine to me, I am having problem with understanding how it settles with the fact that algebraic sets are isomorphic if and only if they have isomorphic coordinate ring. Note that $I(V_1)=\langle(x-1)\cdots(x-4)\rangle$ and $I(V_2)=\langle(x-1)(x+1)(x-i)(x+2i)\rangle$ so by CRT both coordinate rings are isomorphic - as rings - to $\mathbb{C}^4$. Also it seems to me that the isomorphism preserves $\mathbb{C}$ (I thought maybe this was the problem, at first sight)
How do we settle this "contradiction"?