In general I'd say that for any dataset, the ideal visualisation depends on the features and meaning of the data, but almost always some combination of geometry, colour, shape and size can capture all the dimensions. Beyond a small number of categories, colour, shape and size are much less visually informative than geometry. Most often creative use of geometry on different scales is the best first stop for high-dimensional data, for example using faceting (like with ggplot2's facet capability: http://ggplot2.tidyverse.org/reference/facet_grid.html). Simulating 3D is almost never required or optimal.
In general I'd say that for any dataset, the ideal visualisation depends on the features and meaning of the data, but almost always some combination of geometry, colour, shape and size can capture all the dimensions. Beyond a small number of categories, colour, shape and size are much less visually informative than geometry. Most often creative use of geometry on different scales is the best first stop for high-dimensional data, for example using faceting (like with ggplot2's facet capability: http://ggplot2.tidyverse.org/reference/facet_grid.html). Simulating 3D is almost never required or optimal.