Dear Fabien,
Probably I would say that the complexity density of a given pattern will correspond to a certain 'degree' of consciousness. The most 'dense' or complex patterns exist in the brain where we find the highest degree of consciousness.
You agree that a consciousness field is not unreasonable and ask whether this field is already described by our physical theories or whether one needs to add a new one. Charmers thought that we needed a new one and thought that physics 'left no room' for a new field.
In 2006, when I was lead to the idea of a new field, I asked myself how this field could interact with matter. If I thought 'raise my arm' I wondered how the thought actually exerted any force on matter to start the bio-chemical-mechanical process. It took less than an hour to derive a formula for the force of a consciousness field on matter, based on a change in the local field, that was analogous to the electromagnetic force on charge. Similarly, the motion of mass induced a change in the local field, thus inciting awareness of the moving matter. We don't really want to be made aware of matter that isn't changing with respect to us.
It actually took a while for me to realize that the equation I had worked out thinking the problem through in all it it's aspects was actually written down in 1885 by Oliver Heaviside based on his formulation of gravitational theory in analogy with Maxwell's electrodynamics. The more I analyzed the situation, the more every aspect fit together.
In other words, I did not sit down one day and think, "maybe gravity is the consciousness field'. Instead I worked out the simplest equation that exhibited all the properties the consciousness that I thought consciousness must have and then found out that the equations described the gravito-magnetic field of Heaviside that also are the 'weak field' equations derived from Einstein's general relativistic field equations. In other words, I was dragged kicking and screaming to the realization that gravitomagnetism fills the bill perfectly.
Also in 2006 Martin Tajmar measured this C-field in the lab and then 2011 Gravity Probe B detected this field. Eventually, after everything fit perfectly in place, I accepted this ideas, and it has provided the most comprehensive understanding of consciousness that I have come across.
Along the way I realized that physicists, always projecting structure on the world and thinking that this actually describes the world, had misunderstood the 'weak field' equations of relativity. To simplify the non-linear field equations they simply linearized the equations to describe the 'weak field'. Since the equations are no longer self-interactive, they believed the field is no longer self-interacting. This is foolishness. Changing the equations to simplify the calculations does not change the nature of the field. A self-interactive field remains self-interactive. It only means that one must iterate to restore self-interaction to the calculations. Also significant is that it is not mass in the equations but mass density. Physicists again foolishly think that the gravitational field is only significant for large masses. False -- it is density that drives the gravitomagnetic field circulation, hence electrons and atoms induce changes in the local field.
The book I wrote describing this theory of consciousness is "Geneman's World", ISBN-13: 978-0-9791765-5-5, in 2008. My first FQXi essay in 2009 was on the Physics of Consciousness but only ten years ago it was not cool to talk about consciousness in physics. I am quite pleased to see that this topic is now 'respectable'. Believe me, it wasn't.
Warmest regards,
Edwin Eugene Klingman