In an abridged chapter of his recent book Modes of Sentience (2021), University of Exeter philosopher of mind, Peter Sjöstedt-Hughes, argues that higher spatial dimensions might hold the key to the hard problem of consciousness:” He is a fan of the More–Broad–Smithies theory of consciousness:
The word tesseract was coined by the aforementioned mathematician and author Charles Howard Hinton,  whose work on the fourth dimension can be used to our ends. In his essay of 1880, ‘What is the fourth dimension?’ – published four years prior to the related book Flatland by Edwin A. Abbott – Hinton employs analogy to lower dimensional worlds to elucidate a speculated four-dimensional world. I shall briefly explain it, then connect this four dimensional world to the n-dimensional world of Broad and Smythies, so to entertain a theory of the relation between space and sentience. Note that by four dimensions, we are speaking of four spatial dimensions, not a fourth temporal dimension in addition to three spatial dimensions. Peter Sjöstedt-Hughes, “Consciousness and higher spatial dimensions” at iai News (April 14, 2022)
Here’s Flatland (1884), an exercise in the mathematical imagination.
Not clear that a fourth spatial dimension would really give us many insights into consciousness (probably not a material thing) but we may as well keep thinking.
You may also wish to read: Neuroscientists: The Hard Problem of consciousness isn’t so hard! Damasio and Seth tell Nautilus that materialist explanations will eventually crack consciousness, as they have cracked everything else. If consciousness remains unexplained by materialist theories, we are entitled to conclude, at least provisionally, that it is not a material phenomenon.