I think I'll write a blog post at some point explaining my process for visualizing 4D. Hopefully it should make it clearer what I mean.
I think most existing resources on the topic go about it in a way that makes it hard to build up a proper intuition. They start by assuming that humans can visualize 3D and then try to extend that one dimension higher. But humans can't actually visualize 3D, only 2D. We combine multiple different 2D perspectives together to "fake" an understanding of 3D. Our vision is also only stereoscopic 2D, not true 3D.
If you take a similar approach with 4D, trying to project directly from 4D to 2D instead of going through 3D as an intermediate step, it's harder to visualize at first but better in the long run for really understanding it.
Although I (and I assume Patrick above) were joking, I will back you up here- having studied and worked with physicists. Some people can visualize, through practice and psychological “tricks” higher dimensional spaces- and different people can do this to different levels. Some people can instantly intuit approximate geometric solutions to high dimensional problems without working out the math- so I know they are either visualizing it, or doing something similar. I can generally visualize 4D with some “tricks” that compress the problem. For example if you can imagine varying configurations of the same 3D space, that itself is essentially at least a non continuous 4th dimension. If you can imagine that evolving into new configurations, you are starting to visualize or at least imagine 5 dimensions. Some people appear to be able to so this for ~8 dimensions.
Depth is the third dimension for our brains. You can argue it is faked as we don't gather the depth directly, but it is still something you intuitively understand and can visualize clearly. You can rotate an object mentally, fill in the gaps, etc. We may tend to store that data in 2D, but spatial reasoning is very much achievable in 3D. We do it all the time.
While I don't dispute your method works for your purposes, would you say it allows you to visualize more than a single 4D shape side by side? What about interlocking shapes? Can you place this shape in an arbitrary 4D space among others and describe its relative position?