I think that we try to use higher-than-3 dimensional information all the time. Thinking in N-dimensions just means trying to think about N independent variables at the same time. I think that higher-dimensional visual phenomena probably come from the parts of our brain that categorize objects and ideas in N-dimensional spaces. For example, there are many different effects that drugs can produce: Euphoria, abstract thoughts, physical energy, vasoconstriction, immersiveness. If we were to give each of these effects an axis, we would have a 5D graph, and we could then take all drugs and we could put them somewhere in this 5D graph depending on how much of each effect they exhibit. So if you're trying to categorize things based on attributes in this way, then you're trying to think in N-dimensional space, and how we hold this ND space in our minds and how we position ideas and concepts on this space can be very interesting. If we try to visualize this ND space in our heads, we can probably see all sorts of amazing patterns between objects on that space, and I would say that the visuals we see while using psychedelics like DMT are caused by the attempts of our subconscious to visually categorize nonvisual information such as information from other senses as well as information in the form of thoughts and concepts.
Every day I am thankful that I was introduced to psychedelic drugs.