  We give a graphical theory of integral indefinite binary Hamiltonian forms
$f$ analogous to the one by Conway for binary quadratic forms and the one of
Bestvina-Savin for binary Hermitian forms. Given a maximal order $\mathcal O$
in a definite quaternion algebra over $\mathbb Q$, we define the waterworld of
$f$, analogous to Conway's river and Bestvina-Savin's ocean, and use it to give
a combinatorial description of the values of $f$ on $\mathcal O\times\mathcal
O$. We use an appropriate normalisation of Busemann distances to the cusps
(with an algebraic description given in an independent appendix), and the
$\operatorname{SL}_2(\mathcal O)$-equivariant Ford-Voronoi cellulation of the
real hyperbolic $5$-space.
