I think it could be a nice detail if it somehow could be shown if a character had completed the game. Maybe a few lines could be added to the character selection screen to indicate how experienced a character is.

So, if a character had completed the game on casual, something like “Dungeon breeze of the first degree” could appear under the name, and if the character had completed on normal three times, it could say: “Dungeon crawler of the third degree”. For hardcore, “Dungeons master of the…” and for Nightmare, “Dungeon god of the…”   

Just to illustrate the idea, the character selection screen could appear something like this:

Exemplus

J

Lvl 21

Dungeon Breeze of the first degree

Dungeon Crawler of the third degree

Dungeon Master of the second degree

Dungeon God of the first degree