Suppose that I have a disk of radius $r$ around some complex $\alpha\in\mathbb{C}$:
How would one find a complex number $g$ in that disk besides $\alpha$ such that $\mathrm{Re}(g)\in\mathbb{Q}$ and $\mathrm{Im}(g)\in\mathbb{Q}$ and the denominators of $\mathrm{Re}(g)$ and $\mathrm{Im}(g)$ are the smallest possible?
If we have integers $a,b,c,d,p,q,r,s$ with $ad-bc=ps-qr=-1$, then any rational complex in the rectangle with diagonal vertices $\frac ab+i\frac pq$ and $\frac cd+i\frac rs$ has denominators $\ge b+d$ (real part) and $\ge q+s$ (imaginary part). Youcould start with an integer lattice rectangle around $\alpha$ (i.e. $b=d=q=s=1$, $a=\lfloor \Re\alpha\rfloor$, $p=\lfloor \Im\alpha\rfloor$, $c=a+1$, $r=p+1$. Then repeatedly split the rectangle along Farey sums, i.e take $\frac{a+c}{b+1}$ or $\frac{p+r}{q+s}$ (whichever has the smaller denominator), split the rectangle along the corresponding (horizontal or vertical) line and continue with the one containing $\alpha$ until one of the vertices is inside the circle. This gives you the only (or possibly only two) rational complex in that circle with such small denominator, except that possibly the next step gives you equally small denominators in the other coordinate.