$\newcommand\C{\mathbb{C}}$
In the Zariski topology on any quasiprojective variety over $\C$, curves are a distinguished class. They are minimal elements of the set of infinite closed subsets. (In fact, by this type of construction, you can define closed subvarieties in general, and recognize the dimension of a closed subvariety.) Then you can look at fibrations of a quasiprojective variety by curves, in particular fibrations of either $C^2$ or $\C^2 \setminus \{0\}$ by curves. In $\C^2$, there are many pairs of fibrations with the property that every pair of fibers meets in exactly one point. For instance, you can take the fibrations by any two families of parallel lines. The question is delicate because there are many algebraic automorphisms of $\C^2$, and therefore many Zariski self-homeomorphisms. (In fact the Jacobian conjecture is about algebraic automorphisms of $\C^2$.)

I don't think that this is possible in $\C^2 \setminus \{0\}$. I'm going to handwave some, but I think that it works. First of all I think that a fibration by curves has to be an algebraic family, only with its parameterization erased. Suppose that you have two fibrations $F$ and $G$ of $\C^2 \setminus \{0\}$ such that every pairs of fibers meets at one point. Then on each side, 0 is in the closure of every fiber. It's either that or it's in the closure of finitely many fibers. If it were in the closure of finitely many fibers, you would get two families of curves $F'$ and $G'$ in $\C^2$ such that the intersections leap in isolated places from 1 to 2, which is not possible. (The intersection cardinality is lower semicontinuous where it is finite.) On the other hand if every fiber of $F$ approaches 0, then I think that $F$ is a projective family of affine curves, I guess a Riemann sphere of curves. So then a fiber in $G$, which is not projective, would have an algebraic bijection with the projective parameter space of $F$, which is also not possible.

`$\mathbb{C}^2$`

(resp.`$\mathbb{C}^2\setminus\{(0,0)\}$`

) with a face connecting certain curves when they have a nonempty intersection — are these two structures isomorphic? It may not be simpler that way, but it puts the emphasis on curves, and also suggests looking at it from the model-theoretic point of view: the back-and-forth argument can be phrased by saying in terms of an Ehrenfeucht-Fraïssé game between these structures. (And I'm running out of space.) – Gro-Tsen Oct 21 '11 at 21:53