I was wondering what would happen if we wanted to do "complex-like" analysis but, instead, of $\mathbb{C}$, we would use the simplest (in terms of inclusion) characteristic $0$ algebraically closed field, the field of algebraic numbers $\overline{\mathbb{Q}}$.
$\overline{\mathbb{Q}}$ is dense in $\mathbb{C}$ and it can inherit $\mathbb{C}$ topology (sub-question that it raises: can we define this topology without making reference to $\mathbb{C}$, for instance by "extending" the topology on $\mathbb{Q}$?).
This topology allows use to define things like limits, derivatives and $\overline{\mathbb{Q}}$-holomorphic functions.
Question: Do these functions have some nice properties?
(too long for a comment)
A problem is that $\overline{\Bbb Q}$ is not closed topologically, which means that there are convergent sequences in $\overline{\Bbb Q}$ whose limit is not in $\overline{\Bbb Q}$. You can of course include the limits of all convergent sequences, but then you will just get $\Bbb R$.
And the reason one wants the limits of sequences be included in the set is that derivatives are usually defined by taking limits.
It can be done without limits in an axiomatic approach, though:
Let $F=\{f \mid f: \overline{\Bbb Q}\to\overline{\Bbb Q}\}$ be all mappings. Suppose that on a subset $F^*$ of $F$ there exists a function(al) $D:F^* \to F$ such that:
$$\begin{array}{rll} D(f+g) &=~~ D(f)+ D(g) & \quad \text{ for all }f, g\in F^* \\ D(a\cdot f) &=~~ a\cdot D(f) &\quad\text{ for all } a\in \overline{\Bbb Q}\text{ and }f \in F^* \\ D(f\cdot g) &=~~ D(f)\cdot g + f\cdot D(g) & \quad\text{ for all } f, g\in F^* \\ \end{array}$$
The first two requirements state that $D$ is a linear operator, and the third one is the Leibniz rule.
The question is then whether there are non-trivial $D$'s that are interesting enough to start calculus, and that are not already treated as part of (non-standard) analysis.