How can I remove a stuck liquor decanter stopper?
I would recommend not immersing the decanter entirely, but placing it up to the neck in hot water.  This will expand the glass of the decanter slightly, but not the stopper.  That might be enough to let you coax it out.

The shape of your stopper is also convenient for making a little noose for it out of cord/thin rope, to allow you to get some extra purchase on it, to help you prise it out.  So you could hang the decanter and stopper from a beam or similar, and use your weight to pull down on it.  On reflection, and in light of Doug's comment, I'm not convinced this is great advice and should be followed only with serious consideration of the effect on you and the decanter, should the stopper rapidly become unstuck!

As Wayfaring Stranger said, a gentle tap around the sides should help, but be VERY careful not to break the glass, or scratch it. Something not too heavy would be best, and plastic or wooden would avoid scratching - or cover a small hammer with a cloth.