This point has been argued in several discussion threads I've seen. Some make the point that live neurons imply life, while others say the preparation of the squid includes removal of the brain before serving. Such a condition would be incompatible with life (except for U.S. Congressmen).
Miss Cellania has made an intersting (and I think valid) point that the reason for presenting the squid in this fashion is that it provides evidence to the diner that that the food is fresh.
In any case, dead or alive, I would warn potentially squeamish readers that the video is at least grotesque, if not gruesome. I was going to label it "NSFL" (Not Safe for Lunch) -- until I realized that it actually IS someone's lunch...
(The video posted at A London Salmagundi and at Neatorama has been taken down; this one may not last long. If it goes, try searching YouTube for Odori-don.)