There is no such thing as “splitting an action” in the Savage Worlds rules. There is however the ability to take multiple actions in a turn.
Characters may perform multiple actions such as Intimidating someone while blasting away
with a shotgun, running and Fighting, attacking with a weapon in each hand, and so on. […]
Each additional action attempted in a round subtracts 2 from all the hero’s rolls. If an adventurer wants to fire a gun with one hand and slice at an adjacent foe with a sword in the other, for instance, he subtracts 2 from both rolls.
Savage Worlds Deluxe, Chapter 3: Game Rules
What the quickness power does (as printed in Deadlands Reloaded) is grant an additional action without a penalty. So instead of both actions being at –2, both actions are without penalty (or alternatively, you can do three actions at –2, four at –4, etc). However, you still can’t do the same action twice (except for having a separate weapon in the other hand).
Note that the version printed in Deadlands Reloaded matches the version in Savage Worlds Explorer’s Edition (2007). Savage Worlds Deluxe (2011) changed the quickness power to adding a whole entire turn, thus avoiding the ambiguity.
This power grants incredible swiftness to the recipient. With success the target has two separate turns per round on his action card instead of the usual one. Each turn is handled independently with its own actions, but the character must resolve one turn entirely before beginning the second.
If you are playing with Savage Worlds Deluxe as your core rules, then note that the official Deadlands Deluxe-ified PDF supplement says to use those rules instead of the ones in the Player's Guide:
The power descriptions in the Deadlands Player’s Guide are superseded by the Savage Worlds Deluxe versions (but Trappings are unchanged).
I tend to agree that quickness is overpowered, especially in its Deluxe version, but that’s how the rules work.