Actually a 4-dimensional cube would be just an everyday cube. The 4th dimension being time and orthogonal to the space dimensions.
If it exists for any amount of time then it is 4-dimensional.
There are two worldviews on the subject. One sees three spatial dimensions evolving over time.
The other sees four dimensions, one of which is time - x,y,z and t for time.
The first view allows no additional temporal dimensions, the second does, e.g. Vafa's F-theory and Wesson's 5 Dimensional Relativity, 2 Times.
The second view is also expressed in Max Tegmark's Level IV Parallel Universe:
The Platonic paradigm raises the question of why the universe is the way it is. To an Aristotelian, this is a meaningless question: The universe just is. But a Platonist cannot help but wonder why it could not have been different. If the universe is inherently mathematical, then why was only one of the many mathematical structures singled out to describe a universe? A fundamental asymmetry appears to be built into the very heart of reality.
Tegmark, Max, Parallel Universes, Scientific American, May, 2003