  We describe a family of compactifications of the space of Bridgeland
stability conditions of any triangulated category following earlier work by
Bapat, Deopurkar, and Licata. We particularly consider the case of the
2-Calabi--Yau category of the $A_2$ quiver. The compactification is the closure
of an embedding (depending on $q$) of the stability space into an
infinite-dimensional projective space.
  In the $A_2$ case, the three-strand braid group $B_3$ acts on this closure.
We describe two distinguished braid group orbits in the boundary, points of
which can be identified with certain rational functions in $q$. Points in one
of the orbits are exactly the $q$-deformed rational numbers recently introduced
by Morier-Genoud and Ovsienko, while the other orbit gives a new
$q$-deformation of the rational numbers. Specialising $q$ to a positive real
number, we obtain a complete description of the boundary of the
compactification.
