bypass the factory alarm first off. process is on
www.clarks-garage.com
If it's not the factory alarm and an aftermarket one, post back. Most install shops put starter kill relays in.
most alarms will also fault back to whatever they were doing before they lost power so that would be normal and correct.
Is it the normal car horn or does it sound like a different kind of horn? That can also tell us if it's factory or aftermarket.