Nivs..... There may be no option,but to carry your passport if the vote goes for independence.
Part of Alex's aim is to re-join the EU, so signing up to the Euro is mandatory for ALL new entrants - non negotiable. (Which he conveniently forgets). Plus a whole host of other conditions, some negotiable, some not, including Schengen. Hence it may well be necessary to carry your passport, or national ID card to fly between the two.
I don't claim to be an expert on all the implications of the Schengen Agreement on border controls, but I believe that there are different, probably bi-lateral, arrangements at land borders than operate at air or sea ports. So it's possible that there may be no need for a passport to drive across the border, but it will be needed to fly.
The corollary is also that a yes vote might well force the issue for a UK-wide introduction of ID cards.