I’ve thought about this A LOT over the years - and while it's generally thought that we could never possibly imagine what it would be like - I disagree!
Your eyes are really TWO dimensional - like a camera. The retinas of our eyes are essentially flat surfaces (well, they curve a bit - but they are topologically 2D surfaces).
We mostly perceive the third dimension because we have TWO eyes - so we see the two 2D images and our brains infer distance from the small differences between them.
The way that three dimensions get squished into two is with “perspective”. Light travels into our eyes through a small hole and is focussed to make an image.
So - what happens if we were magically dropped into a 4 dimensional world?
Well, if we still have two-dimensional retinas and only two eyes - then you'd say that the fourth dimension would be “projected” into our 2D world in much the same way as our 3D world is - so it would look just like those 2D pictures you see if you google “tesseract” (a tesseract is a 4-dimensional cube):
This is a tanslucent tesseract that's NOT changing shape - it's just rotating around some fourth dimensional axis. It also has perspective in the 4th dimension as well as the 3rd. This isn't some kind of bad visualization of what we'd see. This is EXACTLY what we'd see.
Yeah - it’s a bit weird - but it hardly says “Four dimensional”.
Well, our two eyes are separated by some distance and direction in 3D space - so the two 3D images that get squished into 2D and then our brains use the small differences in the images due to their positions to extrapolate a distance in the 3rd dimension. But this distance is measured at right angles to the line between our eyes.
In an X,Y,Z world (Z is up!) then when we’re standing upright - our eyes differ in their X and Y coordinates - but they’re at roughly the same Z height - so the 3rd dimension distance is the distance in the direction we happen to be looking.
So if we’re dropped into a 4D (X,Y,Z,W) world - where do our eyes end up? Does the line between our two eyes remain strictly in three dimensions - with W constant? Or are we somehow able to turn our heads to look towards the extra dimension? I presume not - we don't have the right W-direction muscles to do that...but perhaps some external device could turn us around in that extra direction.
It actually wouldn't make much difference to what we'd see - we'd just get the distance to the object at right angles to the line between our eyes - and it would still seem to us as if we were living in three dimensions...the two dimensions of our retinas - plus the third dimension we'd get from comparing the two images.
In fact, the only possible way for us to get a feel for the two distances that aren't on our retina surfaces would be if we had a third eye! With a third eye - placed in the middle of our foreheads - we'd be able to turn our heads to place one pairing at right angles to the 3rd dimension axis and the other pairing in the 4th dimension - and what we'd see would be...well, none of us have three eyes - or the brain circuitry to decode them. So all bets are off. We can't tell what that would be like.
But with only two eyes - the answer is simple...nothing very special.
But there is a MUCH bigger problem...
We can see images clearly because light only enters our eyes through that little pupil hole and is focussed onto the retina with a 3D lens. But that hole is a three dimensional hole into a three dimensional spherical eyeball.
Instead of thinking how a 3D eyeball would work in a 4D world, let's do the 2D analogy here. Let's imagine how a 2D eyeball would work in a 3D world!
Draw a circle with a small gap in it on a sheet of paper. That’s a 2D eyeball - it's a circle with a gap in it to let the light in. In 2D, the light goes through the gap in the circle and is focussed onto the opposite side of the circle where the retina is.
But take that 2D eye out into our 3D world and there is a problem. Light can shine from somewhere outside the plane of the paper onto the imaginary 2D retina! So light from almost every direction in the 3D world floods into the eyeball without being focussed by the little 2D lens!
A 2D being would be blinded by unfocussed light coming from the 3D dimension - they’d be unable to focus an image at all.
The same exact thing would happen if our 3D eyes were transported into a 4D world - light would be able to flood into the interior of our eyeballs because a sphere isn't a fully enclosed to the 4th dimension - just as the 2D circle was flooded by light from our 3rd dimension.
We'd be completely blind...we'd probably get some kind of impression of whether it's dark or light - but that would be about it. Unless the light was really dim, we'd be painfully dazzled with no way to close our eyes or shield them with our hands. Very quickly, our vision would be destroyed forever.
String theory says that the world must have at least 11 dimensions (as many as 26 dimensions in some versions of it). But we've just said that if there were more than 3 dimensions then we'd all be blinded. So doesn't this disprove string theory?
Well, not really. The string theorists propose that all but three of those dimensions are "small" so like an ant moving around on a donut (Eeuwww!") there is the "long" dimension around the outside of the donut, and the "short" dimension as you go through the hole in the middle and back around the outside.
If our three dimensions were all VASTLY long (much bigger than the visible universe) but the remaining 8 dimensions were less than a picometer across and 'wrapped around' - then there wouldn't be so much scope for light to travel diagonally in those directions and sneakily bypass our eyeballs to dazzle us.
That might still work.
So there are really two possibilities:
If we (with our 3D eyes) were suddenly dropped into a 4D world, then all we’d see would be a uniform sea of the exact same color. We’d be more or less blind.
We’d see nothing whatever.
If we were propelled into the 4D world with three proper hyper-spherical 4D eyes and 3D retinas - then our brains have not evolved or learned to see with those eyes. We’d have to have different brains - or at the very least to grow up from birth immersed in a 4D world.
Because you and I are not those people - we can’t possibly imagine what the world would look like to them.