Geometrically, the Hopf map is a continuous map from $S^3$ onto $S^2$, with fibres $S^1$. It can also be viewed as linking complex with real geometry, in some sense. Indeed, $S^3$ can be thought of sitting inside $\mathbb{C}^2$, and the domain of the Hopf map can easily be extended to give a continuous map from $\mathbb{C}^2 \setminus \{0\}$ onto $S^2$, with $\mathbb{C}^*$ fibres.
Let us say that instead of $\mathbb{R}$, we had some field $k$, which could for instance be finite. Is there an analogue of the Hopf map for such a field $k$, under perhaps some conditions? One way to make it more precise is this. Is there a field extension $L$ of $k$, such that there exists (under some conditions) a morphism from the affine 2-space over $L$ minus the origin onto the 2-sphere over $k$, and sharing as many properties as possible with the classical Hopf map?