Gotta love a wikipedia article that includes the sentence, “Gently push a piece of the tube containing the intersection out of the original three dimensional space.” 
Because as mentioned above it is (ignoring the semantics, ce n’est pas une pipe), what you’re seeing is not a four dimensional cube (or hypercube), but its shadow. Compare to how a 3D object can cast a 2D shadow on a plane. Our space is equivalent to a plane for a 4D viewer.
And what’s so interesting about the shadow of the hypercube? Rotate it in 4D and it turns itself inside out.
To really address the OP, it seems to me that there’s a fundamental question that never gets a straight answer in these threads: Is there really a fourth dimension to be perceived? The Flatland book mentioned above contains a vignette where the 2D square being gets shoved off of his plane of existence and flies around momentarily in the third dimension, such that he’s able to look “down” and see inside his neighbors’ houses. Yes, yes, that was all written for illustrative purposes. But could something like that really happen to one of us three-dimensional creatures? Or is it firmly established that the whole fourth dimension business is just a theoretical construct (like, I suppose, imaginary numbers) that could never be encountered in a physical way?
And the first person who comes back with “you do move through a fourth dimension, it’s called ‘time’” – gets smacked with a trout, because you know what I’m talking about.
Imaginary numbers are just as much a theoretical construct as real numbers (or more accurately, real numbers are just as theoretical as imaginary numbers). I don’t know all that much about physics, but I believe some theories posit that the universe is n-dimensional (with n > 3 and even n > 4), but then again only a certain number of dimensions are supposed to be space-like. Someone who knows more about this than I will be along shortly.
If we are viewing the shadow of the 4D cube, why does it have to be moving, in our perspective?
A cube would not appear "moving’ to someone in 2D land.
M-Theory posits the existence of 11 spatial dimensions.
However, the 4th - 11th dimensions are all closed loops.
And, I know this doesn’t help at all.
That’s just to show the structure more clearly as it rotates.
Yes - imagine the 2D shadow of a 3D cube as it rotates - the shape of the shadow changes. The diagram is a 2D representation of the 3D “shadow” that you would see in our dimension if the tesseract was rotating in four dimensions. The “shadow” itself is not rotating, it is physically changing shape, just the same as the shadow of a rotating cube actually changes shape as the cube rotates.
Koxinga, if you want to really blow your mind, it’s not even certain that there are three spatial dimensions. There are theories (with some experimental evidence behind them, even) that what we think of as a three-dimensional universe is actually just an emergent property closely analogous to a hologram arising from a fundamentally “real” two-dimensional space.
If you want to own your own 3d representation of a 4d object, you can find it here.
From ACME Klein bottles.
There is lots of entertaining reading there.
Personally, I like the volume decals the include with the Erlenmyer-style bottles. They’re labeled
— 000 ml
— 00 ml
— 0 ml
— 0.0 ml
— 0.00 ml
As Hypnagogic Jerk notes, all mathematics is theoretical (or, perhaps better put, abstract); imaginary numbers are no different from anything else in this respect. But even moreso, you are deeply, intimately familiar with physical realizations of imaginary numbers already (although those stupid names “imaginary” and “real” numbers get in the way of people realizing this); you use this familiarity every time you take two left turns to achieve a U-turn. Complex numbers are just combinations of scaling and rotation, as explained in this post and further discussed in this thread.
I have no idea what a 4D “spectrum” is. What stops you is that the universe has 3 spatial dimensions, not 4. (Time is in some sense a fourth dimension, but not spatial). So there is no fourth dimension to see.
Some people claim to be able to visualize in 4 dimensions, but most people can’t, probably because our brains evolved in a 3D universe.