If consciousness is not embodied that would imply metaphysics and body/mind dualism, e.i. something countrary what one might term the scientific world view.
"what one might term the scientific world view" is great and very practical but not perfect. There certainly are legitimate questions it probably has no way to address.
I feel pretty skeptical about exploring the nature of consciousness without employing techniques available in Buddhism, psychonetics[1] or something like that. These are fundamentally unscientific (although not imaginary - execution of the practices can be easily detected with EEG) but can at least let us observe the object of the study with at least some basic degree of objectivity, reason about it and do something useful with it.
My best guess is qualia and consciousness are like an EM field, something that exists and has some relationship with known physical entities but is distinct at the same time. It may be amplified by, say, some computational structure, but exists independent of it.
At one time I had convinced myself that qualia were related to time, in the sense that it was the "substantive" manifestation of time as a dimension, but now I'm not sure what I was thinking.
I suspect in the end consciousness will end up in the domain of physics rather than neurobiology and psychology alone.