For the purpose of this questions, let's assume my crew has solved all those nasty problems of survivability inside the field, not unleashing a devastating blast of particles at their destination, and energy requirements.
What would the solar system look like to someone inside the ship while the Drive was active? If they for some reason decided to look at the Sun through a window, what would they see travelling away from it? Would they even be able to see anything at all?
According to wikipedia (which is more than sufficient for a layman-level discussion, hard-science or otherwise):
The Alcubierre drive [...] is a speculative idea [...] by which a spacecraft could achieve apparent faster-than-light travel [...].
Rather than exceeding the speed of light within a local reference frame, a spacecraft would traverse distances by contracting space in front of it and expanding space behind it, resulting in effective faster-than-light travel.
When I was getting my degree in physics (almost two decades ago), I wrote a paper on the visual effects of near-luminous travel. While this isn't the same thing, there are two similar circumstances that immediately leap out at me - both dealing with changes in angles: shape and color.
First, the shape of things changes. The more the space ahead of you contracts (and the space behind you expands), the 'smaller' things ahead of you look and the 'bigger' the things behind you appear. Angles of objects slightly off-center ahead contract until - at some limit - they appear directly ahead; the more contraction of space, the tighter the new angle. For instance, if you are heading straight for a very large object (a galaxy, for instance, so we don't run into it right away), and it forms a rough circle directly ahead, the more the space contracts, the smaller that circle will be as the center remains unchanged by the edges contract in. A similar, but reversed, effect happens to objects behind you: they expand the 'cone' from you to them. A circular galaxy directly behind you will expand as the center remains unchanged by the edges all widen. At extremes, things directly behind you will seem to be 'swallowing' you as the angles pass 90°...
Second, the color of things changes, too. The very wavelength of light undergoes the same contraction (ahead) and expansion (behind). Wavelength of light affects, among other things, color - so things ahead of you will tend to "blueshift" and things behind you will tend to "redshift." At smaller 'levels' of contraction and expansion, this would just make things look weird... but the electromagnetic spectrum is more than just the visual spectrum. At extremes, things behind you will shift beyond red into the infrared then micro- and radio waves, and things ahead of you will shift beyond blue into the ultraviolet then into ionizing radiation. Hopefully, your culture that has invented a working FTL drive understands that they'll need radiation shielding...
For near-luminous travel, these two effects don't start becoming really interesting until you hit something like 0.99999c, though they'll be visible to the naked eye long before that. The question then, becomes, how much "contracting" and "expanding" of space you're doing in your drive. (I'd provide equations, but I'd have to dig them out of a box in the basement - and while they don't require anything too nasty math-wise, they do involve four-dimensional vector matrices, which gives mathphobes nightmares. I haven't touched this stuff in close to two decades, so I'd rather not try to wrap my head around the actual physics - just the high-level layman descriptions.)
The answer to this question is easy in description...
Depending on whether you are looking in front or in back of you, you are confronted with either a field of white or a field of black.
The reason for this is, if I remember correctly, as you move forward you are running into every bit of light there is at speed that the little light that is there gets built up, multiplied and shifted into the visible light spectrum (and beyond) making the white field while behind you no light can catch up to you which makes the black field.