Sleep experience, especially dreaming, lies near the heart of The Notion Club Papers (NCPs). One aspect of this is that there are multiple references to the idea that the dream world is a realm of experience which is universal
- in other words, dreaming
is a single, vast domain - with distinctive qualities, different from the waking state - that is potentially accessible by all people.
For example, through the course of the the NCPs, club members begin to dream of the same Numenorean material; and eventually Lowdham and Jeremy come to meet each other in dreams, and share a dream of a sea voyage in Anglo Saxon times.
Prior to this, Lowdham has learned of two languages in dreams, while Jeremy has had visions. Frankley receives in a dream the long poem about St Brendan (Imram) which also refers to Numenor.
The NCPs open with a description of how Ramer, by a course of practice and training - and with the assistance of Dolbear - becomes able to travel both in time and space in his dreams; visiting other planets, and seeing a from-above vision of the sped-up history of Oxford across many centuries.
In the discussions; club members refer to actual and possible encounters with hostile spirits during dreams - making the NCP dream world reminiscent of the Ancient Egyptian 'underworld' Duat/ Dwat where resided the gods, including malign 'demons' such as Set.
The 'plot' of the (incomplete) NCPs could, indeed, be said to be about how the club learns the shamanic practice of 'lucid dreaming' - that is conscious and purposive dreaming - as a way of contacting and learning-from a 'spirit world' which includes not just objectively-accurate historical data.
Such ideas are quite general in ancient and esoteric cultures; but are specifically similar to the 'Akashic Records' mentioned by various mystics, including Anthroposophy founder Rudolf Steiner. The Inkling Owen Barfield (Jack Lewis's best friend from Oxford days) was perhaps the leading Anthroposophist writer in England, and several other Lewis's close friends were also Anthroposophists: e.g. Cecil Harwood (who became Lewis's literary executor) and and Walter O Field (who was a companion on walking holidays).
The idea of a universal dream realm is also quite a common feature in fantasy fiction; I am currently reading Robert Jordan's vast The Wheel of Time epic (regarded by some as the greatest world-building fantasy since Tolkien) where the dream world of Tel'aran'rhiod has a vital and frequent role in the plot.
The importance of the dream realm is that it is also the mythic realm - and this links it (in broad terms) with Jung's Collective Unconscious. The NCPs assume that dream experiences are potentially real
experiences - with waking-life consequences - as when the storm from The West which destroys Numenor breaks-through to wreak havoc on the modern day British Isles.
The idea is that knowledge may be obtained and communications may happen in dreams that are otherwise inaccessible to the waking state. The challenge for the Notion Club is to become conscious in dreams, to gain some control over the dreams while they are happening, especially so as to direct them - and also to remember and recount to the other club members what has happened.
I strongly doubt whether there was any direct influence of Anthroposophy on Tolkien (despite that one listed member of the Notion Club was the parodically-named Ranulf Stainer!) - but there is an unwitting convergence between the aims of the Notion Club members, and aims of Rudolf Steiner's spiritual exercises; which are supposed to have the side-effect of inducing 'lucid dreaming' with awareness of the dream state, increased recall and some degree of control of dreams.
At any rate - the broad idea of a universal dream world seems to have been one which at least fascinated Tolkien - but most likely was also an idea that he personally believed-in and had personally experienced.