RAW is pretty clear afaik, spells don't automatically end upon the death of the caster unless they their duration is Concentration (as a caster obviously can't concentrate on anything if they are dead). Dominate Person lasts 1 day per caster level, and in addition since it does not have (D) in its duration, Durkon* would have needed to cast dispel magic (albeit auto-succeeding on a dispel check) to break the spell.
That said, the comic isn't following DnD rules to the letter. The above interaction with Dominate Person is (imo) dumb and very counterintuitive.