I am interesting in classifying the subrings of $\mathbb C$ which are discrete with respect to the standard topology (that is, the topology induced by the standard absolute value). Here, I am using the word 'discrete' to mean that every element of the ring is an isolated point. Here is what I know so far: (let $R$ denote such a subring)
-$R$ cannot be a field.
-$R \cap \mathbb Q = \mathbb Z$.
-The unit circle centered at any element of the ring can only contain its center, i.e., for any $b \in R$, $$ \{ a \in R \, \vert \, \left| a - b\right| < 1 \} = \{ b \}. $$
-The unit group of $R$ must lie on the unit circle (centered at the origin).
-Examples include the rational integers, the Gaussian integers, and the Eisenstein integers.
A discrete subgroup of $\mathbb C$ is free of rank either $1$ or $2$.
If $R$ is a discrete subring of $\mathbb C$, then the above implies that it is either an free abelian group of rank at most $2$.
If the rank is $1$, let $x$ be a generator. There is an $n\in\mathbb Z$ such that $nx=1$, so $x=1/n$. If $n$ is not $\pm1$, then the powers of $x$ converge to zero and are all in the ring: since the ring is discrete, this is impossible. We thus see that in this case the ring is simply $\mathbb Z$.
Suppose now that the ring is free as an abelian group of rank $2$. By the same reasoning as above, there cannot bee an element $x\in R$ such that $1=nx$ for some integer $n\neq\pm1$. This implies that there is a basis of $R$ as an abelian group of the form $\{1,u\}$ for some $u$. Now $u^2$ is in the ring, so there are integers $a$ and $b$ such that $u^2=a+bu$. Now $u$ cannot be real (for there are no discrete subgroups of rank $2$ in $\mathbb R$) so the discriminant of the polynomial $X^2-bX-a$ has to be negative: it is therefore irreducible over $\mathbb Q$ and we see that $R$ is contained in the ring of integers of a field of the form $\mathbb Q(\sqrt{-d})$ for some positive integer $d$.
By a well-known result, the non-real quadratic integer rings are of the form $\mathbb Z[\sqrt{D}]$ or $\mathbb Z[(1+\sqrt{D})/2]$, with $D$ a negative square-free integer, according to whether $D$ is congruent to $2$ or $3$ modulo $4$ or to $1$ modulo $4$. The ring $R$ need not be equal to a ring of this form: for example, $\mathbb Z[2i]$; in any case, these gives us the maximal discrete subrings of $\mathbb C$.