Best new franchise? Probably Uncharted.
The first Bioshock is certainly my favorite single game in that list of IPs: The story, the setting, the voice acting... the game is literally interactive art the way it's told and brought to life. But I thought the second game was a cheap cash-in and Infinite just didn't retain that magic from the first game.
The first Uncharted was a decent freshman entry with a solid game engine, nice graphics and fun protagonists. But the charm and polish of Uncharted 2 solidified the IP as something associated with very high-quality going forward and both Uncharted 3 and 4 were great.
So, even though the Bioshock IP has the most potential for an epic, as of right now, I'm going with Indiana Drake for consistency.