Structural dynamics systems can be represented in terms of mass, damping and stiffness matrices. Each of these matrices may be coupled depending on the model complexity, degrees-of-freedom, etc. The mass and stiffness matrices in the assembled equation of motion may be uncoupled using the normal modes for the undamped system. This approach gives real natural frequencies and real mode shapes.
Damping effects can be included in forced response analyses by implicitly assuming that the damping matrix can be diagonalized into modal damping coefficients by the undamped modes. But systems with dashpots in general have damping matrices which cannot be uncoupled in this manner.
The state-space method is useful for modal and forced response analysis of systems with discrete dashpot damping. This approach yields complex natural frequencies and mode shapes, with real and imaginary components.
Here is a paper:
Two-Degree-of-Freedom System, State-Space Method: two_dof_state_space_revC.pdf
– Tom Irvine