It has been a while since I have blogged about the weird and wonderful "rules" that players believe are proper chess rules. The most recent addition to my list came from a schools event I ran last week. One poor player, playing in their first proper chess event, was most upset when he was told he had been checkmated. Wile he accepted that yes, he was in check, and yes, there was no way to get out of check, it still didn't count as he still had his queen on the board. "You can't get checkmated while you still have a queen" was his claim, a claim I had to explain to him, was incorrect.
Nonetheless this "rule" may make a interesting chess variant. Even if you are checkmated with queens on the board, you can still play on, as long as your queen survives. The game might then turn into a battle between the enemy pieces and the queen, while the defending side could be torn between using the queen to lift the mate, or hiding her as far away as possible.
I suspect the overall result would be longer games, but on the other hand, players would be less inclined to swap queens in the opening.