Proponents
William James is given credit for the concept. He was enormously skeptical about using introspection as a technique to understand the stream of consciousness. "The attempt at introspective analysis in these cases is in fact like seizing a spinning top to catch its motion, or trying to turn up the gas quickly enough to see how the darkness looks."
Bernard Baars has developed Global Workspace Theory which bears some resemblance to stream of consciousness.
Read more about this topic: Stream Of Consciousness (psychology)