Oh, cool! Yeah, having now seen them all the Corey version does seem most readable, though I wonder if some kind of 3-D animation when changing viewpoints could add legibility to other... in any case, it's an interesting space to explore, and makes me think about dimensional systems in other genres, like a metroidvania where a dimension might be used for fast-travel or for breaking into secret areas somehow, as a kind of tangentially useful mechanic, rather than the core mechanic of puzzle design. Fun stuff.
We cannot grasp 4 spatial dimensions. It is not even known if they exist. It is proven that in 4 dimensional space a planetary system could not be stable. So for all intents and purposes, ouside mathematics, there exist only 3 dimensions. Also if there were 4 dimensions, why should objects only have 3. If you count time as a dimension, objects sure have a length in those as well.
Per definition an added dimension is a direction perpendicular to the others. It is easiest to imagine as moving in time (althoug that is not a spatial dimension). So if you could move along the non spatial axis to a point where the secret cave entrance is open, you can then use your regular 3d movment to go in and collect the secret.
What about direct 4d movment then. If nothing exists perpendicular that would block your path, you could initiate a movement and while moving shift along the perpendicular axis for a short time to arrive behind the barrier. If nothing there blocks your path, you cannot just shift there and move then, because even the ground you are standing on would not exist, but neither would gravity. Or Air.
A movement ability or device that would just transport you from A to B would be indistinguishable from plain old teleportation, maybe by how long the movement would take. That has been done countless times. Calling it 4d movement adds nothing but the can of worms of side effects. (including people mistaking it for time travel). Like dying from lack of oxygen if your travel takes too long. Travling along a new axis does not make the distance shorter. Yes, yes, in higher dimensions the space could be curved and the distance could be shorter, as explained in those scify stories with wormholes. But why should it be curved just so that your travel through a mountain to reach a secret cave is short. Your ability might do that, but this would be like those secondary super powers. Just call it teleportation. Phased movement. Shadow Step. Ghost realm. Hyperspace, etc.
Oh, I like to ponder such thought constructs. Ever wonder why ghosts can walk through walls but do not fall trough the ground. The answer is of course, "magic". It does not follow regular rules. But if you make your super power be a mechanic, questions arise. Like why that awesome ability is not used in other circumstances. You could of course make up all kinds of additional rules, like ghosts could do it, but they do not think of it. Or you could just stay with the "magic" explanation.
Of course exploring all the side effects of a mechanic can be fun in a plot. But more often leads to situations like in that Star Wars movie, where they finally decided to apply real phyiscs combined with available tech. Ramming a thing with very high speed. That is a break from the usual magic explanation for tech there and leads to the question, if this is possible, why did they not destroy the Death Star(s) with that. Or planets for that matter. Why build a Death Star in the first place, if you can just accell a drone ship to near light speed in the direction of your target.
Oh, btw, the concept of changing view in a puzzle has been seen lots of times. Either you change your char, the view or even both.
The more abstract puzzles like this one here can get away with 4d, but in the usual platformers, that would be hard.
I even remember a time puzzle in that old point&click Day of the Tentacle. You had to change stuff in the past to solve puzzle in the present.
If you can pull it off, a new mechanic can be awesome. Portal comes to mind.