I just won the final battle and defeated Tyranthraxus. The party was automatically teleported out and all that. But, now I'm stuck at the "training" screen (where you can add/remove/train/etc party members). I can view my chars and all but if I try "Begin Adventuring", nothing happens.

I was hoping I'd be able to wander around and maybe fight some more random monsters in the dungeons or what not. As well as ID the magic items I found at the end.

Is this not possible?

Sadly, this is not a bug, but a feature. It does that if you win. I guess they didn't want to bother programming in an 'aftergame'. (Some of the later games do.)

The lack of exploration after the ending in CotAB was disappointing, but at least the game offering a nice ending sequence. Buck Rodgers: Matrix Cubed was the worst, that game just dropped to the MS-DOS prompt after the game ended.

The next game (Secret of the Silver Blades) in the series is much better in this aspect, you can explore as much as you want after the ending, and see the after effects of what happened.

The first Eye of the Beholder does the same. The really weird thing is that CotAB is the only game in the Pools series that doesn't allow "post-game exploring" (maybe Pools of Darkness doesn't either; too long since I've played it). Even PoR allows you to go exploring after you defeat Tyranthraxus and return to the civilized area of Plan. Not that there's much to do, but still.

From what I have read, not only does Pools of Darkness let you explore after beating the game, but there is apparently a bonus dungeon. This is something that, while common in recent JRPGs, was not common back then and is not something you generally see in a WRPG even now.

Yeah, and it's pretty hard.

Death Knights of Krynn also has one.