Providing objective metrics of conscious state is of great interest across multiple research and clinical fields-from neurology to artificial intelligence. Here we approach this challenge by proposing plausible mechanisms for the phenomenon of structured experience. In earlier work, we argued that the experience we call reality is a mental construct derived from information compression. Here we show that algorithmic information theory provides a natural framework to study and quantify consciousness from neurophysiological or neuroimaging data, given the premise that the primary role of the brain is information processing. We take as an axiom that "there is consciousness" and focus on the requirements for structured experience: we hypothesize that the existence and use of compressive models by cognitive systems, e.g. in biological recurrent neural networks, enables and provides the structure to phenomenal experience. Self-awareness is seen to arise naturally (as part of a better model) in cognitive systems interacting bidirectionally with the external world. Furthermore, we argue that by running such models to track data, brains can give rise to apparently complex (entropic but hierarchically organized) data. We compare this theory, named KT for its basis on the mathematical theory of Kolmogorov complexity, to other information-centric theories of consciousness. We then describe methods to study the complexity of the brain's output streams or of brain state as correlates of conscious state: we review methods such as (i) probing the brain through its input streams (e.g. event-related potentials in oddball paradigms or mutual algorithmic information between world and brain), (ii) analyzing spontaneous brain state, (iii) perturbing the brain by non-invasive transcranial stimulation, and (iv) quantifying behavior (e.g. eye movements or body sway).