I don't have any personal experience with either unit but I will say that I've been extremely pleased with my P800 and absolutely thrilled with the P99.
The specs on the Alpine aren't really impressive, I Googled the P88RS II and found some Euro specs but don't see it in the Pioneer US website so not sure if this is the Euro version of the P880 or not... They both look the same except for the all-black faceplate on the Euro version vs. the brushed trim on the US P880.
If it is I'd say it's a no-brainer, you get 16-band EQ on the Pioneer vs 5-band on the Alpine, you get Burr-Brown DACS and, in my limited experience, the Pioneer auto TA/EQ works really well. Don't have any experience with the Alpine Imprint system so can't offer a comparison.
I've owned high-end Eclipse, Alpine and Pioneer HUs and, IMHO, the Pioneer blows the others away. The Alpine(from specs, not experience) looks like a really nice deck but not in the same class as the Pioneer.
Take my observations with however many grains of salt you wish, I'm no audio junkie. I find something that works for me and stick with it until I find something better. I was absolutely blown away with my P800 and even more impressed with the P99 so I guess I'm officially a Pioneer fan..