A four dimensional space, as I understand it, is just a set of points each defined by four numbers. It's easy to visualize the first three, so then I just imagine the last one as the color of the point. So a 3D hyperplane in a 4D space is just all eight octants with gradually changing color along an axis parallel to some vector which represents the gradient of the 3D hyperplane, and the color is constant along any vector perpendicular to it.
If I want to "look at" a 4D graph, then in my head I just imagine a 3D surface of one uniform color, which I understand to be one of an infinite number of cross-sections of the graph, and then modulate the color and watch the graph slowly and continuously morph as I do (assuming the graph is continuous).
Is this conception misguided? I've always used it and I'm surprised I haven't heard anyone else mention it. Does it generalize? For example, could I imagine a 6D hyperplane in a 7D space in the same way but with a separate coordinate/value for R, G, B, and opacity of the point?
There are several ways to visualize high dimensional data. The clever use of color or (point) density is one of them. Another general one is taking sections, for instance by fixing enough coordinates to achieve a plot in 1D, 2D, or 3D again. Similar to that one can also take projections to a subspace of lower dimension, which achieves the same thing. A really interesting one, and not often mentioned, is by using parallel coordinates, a concept popularized by Alfred Inselberg. In this way of plotting each point in the nD space becomes a line connecting n axes in 2D. Ranking the lines, or using color can additionally add information to make the plot more rich in information and accessible for understanding. This essentially also generalizes to nD data really without problems, and, in my opinion, has some advantages over both sections/projections.