Red dots come in tube types, like Aimpoints, and open reflex like Fastfires. The reflex give you a more open view, with usually just a thin frame around the “window”, and are better IMO for a shotgun, especially if you may take shots on the wing. The FF3 is reasonably priced and excellent, plus the battery can be swapped without removing it from the base. If you’re “accident prone”, they make a tactical mount with protective wings for it.
I have a tiny Bushnell Mini P prism sight on my Henry .410 for this season. Advantages are that it works with or without batteries (illumination off), and the focusing ocular (no fuzzy dot). It allows very precise aiming, almost like a peep sight. The main disadvantage is that it has limited eye-relief like a scope, so needs to be mounted as far back as you can manage. The view is a little tight too, but I tolerate that because it’s so tiny compared to other prisms. The FF3 is equally tiny and lighter, so I may end up getting one for the Henry as well, as soon as I pay off the mortgage on my TSS shells.
