  By analytically continuing the eigenvalue problem of a system of two coupled
harmonic oscillators in the complex coupling constant $g$, we have found a
continuation structure through which the conventional ground state of the
decoupled system is connected to three other lower {\it unconventional} ground
states that describe the different combinations of the two constituent
oscillators, taking all possible spectral phases of these oscillators into
account. In this work we calculate the connecting structures for the higher
excitation states of the system and argue that - in contrast to the four-fold
Riemann surface identified for the ground state - the general structure is
eight-fold instead. Furthermore we show that this structure in principle
remains valid for equal oscillator frequencies as well and comment on the
similarity of the connection structure to that of the single complex harmonic
oscillator.
