I was doing something similar; for one dimension, I imagine a sequence of dots. the 2nd dimension adds a series of new lines of dots, forming a flat sheet, the the third dimension adds new rows of these sheets.

for the fourth dimension, you regress back to the first; now all dimensions before are encapsulated within each dot.

however, in the past I came across a website which allowed you to manipulate a hypercube, rotating it through our familiar dimensions as well as the third it extends into. I found myself utterly unable to predict how rotating the two dimensional image of a 3d representation of a 4d object would alter what was displayed.

ever since then I’ve been curious to learn to envision what such an object would “look” like. these ways of thinking about higher dimensions don’t really shed any light on that.