look, I'm not disputing this second mind you perceive, I'm rather corroborating it. I myself have this superpower of offloading problem solving to an intuitive unconscious co-processor that has given me pretty good solutions (and lots of anxiety, as I have no insight into its progress before it gives me an answer, that may or may not be 42 ;-) but in order to function, it requires me not to focus my conscious mind into demanding tasks. whether that means they're sharing the same computing resources, or just that other demanding tasks would draw the intuitive mind's attention, is not known to me. I also need distraction, say, a TV or radio producing background noise that I don't even notice, to be able to fully focus my conscious mind, as if there was some part of my mind that, if not given this distraction, would disrupt the part that is trying to focus on something. it's truly fascinating.
the fable of the struggle between two wolves inside us all is more supporting evidence of our having more than one mind. but why stop at two? we run strange loops in our minds that stand for our models of others' minds, and some even have multiple personalities, in which multiple separate minds seem to inhabit the same brain. IIRC both hoffstadter and damasio perceive some link between the self-referential nature of the mind, with multiple layers of feedback loops, with this explosion of possibilities and complexities