Let $E_1$ and $E_2$ be elliptic curves over $\mathbb{C}$. A (primary) Kodaira surface is a principal bundle $X \to E_1$ with fibre $E_2$. $X$ is a compact complex surface with trivial canonical bundle and so it has Kodaira dimension $0$. In general $X$ is not algebraic (not even Kaehler because $b_1=3$).
The question is: can this construction be generalised to some other fields $k$ (and some appropriate topology)? If the answer is yes what kind of object would one get?
You could follow Suwa's construction of the Kodaira surfaces from his paper Compact quotients of $C^2$ by affine transformation groups, and define various surfaces which are quotients of $k^2$ by groups of affine transformations of the special form that Suwa arrives at in his paper. I don't know any applications, but it is a very nice paper.
If I remember correctly, every Kodaira surface arises this way, as shown in Inoue, M., Kobayashi, S_ Ochiai, T.: Holomorphic affine connections on compact complex surfaces J. Fac. Sci. Univ. Tokyo 27 (t980)
