I'm with you in finding the notion that consciousness would continue forever terrifying. Just ceasing to be doesn't frighten me. I'm not through enjoying being conscious, so ceasing to be isn't attractive and I do try to avoid it for now, but the scary parts are the unpleasant stuff that so often comes between fully functional and not there any more, and being unable to completely shake off that supposedly comforting notion that my soul might truly be immortal.
Of course, that's on an intellectual level. I still have the awful instinctive fear of death when the car starts to slip on that patch of ice. Our instinctive fear of death is one of those dirty deals we get, inevitably, because of being evolved creatures. The quality that makes us people is being able to consciously transcend instinct. We don't have quite enough of it.