Let's ignore the name "Alcubierre" for a moment
You see, the Alcubierre drive doesn't exist. For all intents and purposes, it's simply a name... a piece of technobabble. We have an idea of how it might operate in really general terms, but without a working model, that's all just more science fiction.
Which is good for you!
Let's address what I believe is your real question. You want to know if it's plausible (aka, "believable") to describe a fourth geometric dimension that our fictional drive can travel through?
It's perfectly satisfactory to throw out time as the forth dimension. After all, the "first three" all describe geometric relationships, why not the fourth? The idea isn't new, a tesseract is to a 3D cube what a 3D cube is to a 2D square. So, we have a geometric 4D universe where there exists a tesseract with 4 equal length legs extending from all 8 verticies.
The question becomes, if you can dtravel through such a unverse, along that 4th leg, would it buy you anything in terms of travel?
The answer is complex, but cool1
Answer #1: No Remember that tesseract? the only way to get from the 3D "inner cube"2 to 3D "outer cube" is by traversing through 4D space. To make it simple, we'll only travel vertex-to-vertex along legs. The shortest distance from any vertex of the inner cube to the closest vertex of the outer cube where the length of all the legs is "x" is 2x. In fact, as we'll see later, this is actually costing you distance and time to traverse the 4th D.
Answer #2: Yes Except that 4D space will allow manipulation that can't be achieved in 3D space. Let's assume a piece of paper represents an infinitely thin 2D plane. Our 2D people want to get from one point to another very distant point. We godlike beings in the 3D universe can easily see that the paper can be bent such that the distance travelled through the 3D universe is very short... even ZERO.... We can bend the paper such that it touches.
Now, this assumes that the 2D plane actually can be bent. We have (and this is REALLY IMPORTANT) not one single example of an actual 2D universe. Not one, nada, zilch. We hypothesize that we can do it because we can easily bend a sheet of paper, but paper isn't actually a 2D object or a 2D universe. In fact, the reality that we do not (or, at least, have not yet) perceived a true 2D universe in any way strongly suggests (but doesn't prove) either (a) there are no other traversible dimensions than the 3rd dimension or (b) No one inhabiting a dimension (2D, 3D, 4D...) can perceive any other dimension.3
And this is OK, because the ability to perceive the 2D universe would mean the ability to manipulate the 2D universe and that would be very godlike and I'm not feeling that today.
Answer #3: Well, no, kinda... yes... but... The next realization from our 3D perspective is that between the top of my head and the soles of my feet is an infinite number of 2D planes. a 2D creature could conceivably travel a distance through 3D space... but the little tyke will end up on another 2D plane. They can pass a distance through 3D space and take an enormous amount of time along a parabolic course to arrive at some point on their own plane (it would always be a longer time due to the shortest-path-between-two-points issue).
In this case we can't bend our universe (the 2D universe) but can travel through the infinite "planar" universes that comprise the 3D universe. In like manner, we travel through the infinite planar 3D universes that make up the 4th D. But it would always take a longer period of time and a longer physical distance.
But the Alcubierre drive is all about bending that piece of paper...
And that's where the magic of fiction comes in. We have this mathematical theory that suggests we can do it. The hint of the possibility makes eyes grow and mouths drool. If we can bend that piece of paper, the only way it can be bent is through a 4th material dimension, just as bending 2D space would need to happen through a 3rd material dimension. (Ignoring Footnote #2 and the ramifications that it's not actually distance that's being manipulated....)
So, ultimately, if you use the Alcubierre drive as your reference, the answer is Yes.
As for whether or not things like magnetism and gravity flow through the 4th dimension, that's a good question. Mathematically they do, but we don't have a shred of evidence. What does it mean to a densizen of 2D land to have a gravitational force in the 3rd direction? It means nothing. What would "gravity" caused by an infinitely thin 2D "mass" mean to a 3D universe? We have no example of anything doing such. And what would be the impact of "gravity" caused by a 4D mass mean to us? So far it means absoltuely nothing at all. All the gravity we can detect is accounted for by 3D objects.
In short, while some tend to treat mathematics as written-in-stone proof, I tend to treat things that only exist mathematically as cool suggestions and hints until actual proof arrives. Violating a mathematical theory before physical proof cannot make your idea suddenly "unrealistic."
1 If HDE 226868 disagrees with anything I've said here, he's right. His left big toe knows more about this than I ever will.
2 Really, this is from the 3D perspective of the Teserract where it's often drawn with an inner and an outer cube and the reader is expacted to deal with the mind-bending problem of the two cubes having the same physical size and volume. It's a bit like thinking about Dr. Who's Tardis and it's often to take it on faith. I'm using this reference simply to make a point and hope I'm not being too confusing in the process.
3 By the way, this is REALLY STRONG EVIDENCE supporting time as the true 4th dimension. We do have evidence that "space-time" is bent by gravity. This isn't a shortening of the geometric distance between two points, but the relativistic perception of the distance travelled due to the bending of space-time. In short, we know space-time can be bent, we can prove it. That kinda disproves the 4th D being geometric, at least in my mind. But that shouldn't stop you from writing your story at all.