Geometry of integral binary hermitian forms

We generalize Conwayʼs approach to integral binary quadratic forms to study integral binary hermitian forms over quadratic imaginary extensions of Q. We show that every indefinite anisotropic form determines a plane (“ocean”) in Mendozaʼs spine associated to the corresponding Bianchi group in the hyperbolic 3-space. The ocean can be used to compute the group of integral transformations preserving the hermitian form.