  The central theme of this paper is the holomorphic spectral theory of the
canonical Laplace operator of the complement $\Omega := \{(z,w) \in
\widehat{\mathbb{C}}^2 \colon z \cdot w \neq 1\}$ of the "complexified unit
circle" $\{(z,w) \in \widehat{\mathbb{C}}^2 \colon z \cdot w = 1\}$. We start
by singling out a distinguished set of holomorphic eigenfunctions on the bidisk
in terms of hypergeometric functions and prove that they provide a spectral
decomposition of every holomorphic eigenfunction on the bidisk. As a second
step, we identify the maximal domains of definition of these eigenfunctions and
show that these maximal domains naturally determine the fine structure of the
eigenspaces. Our main result gives an intrinsic classification of all closed
M\"obius invariant subspaces of eigenspaces of the canonical Laplacian of
$\Omega$. Generalizing foundational prior work of Helgason and Rudin, this
provides a unifying complex analytic framework for the real-analytic eigenvalue
theories of both the hyperbolic and spherical Laplace operators on the open
unit disk resp. the Riemann sphere and, in particular, shows how they are
interrelated with one another.
