If you attempt to conquer an empty battlefield and play Arcane Shift, can you play the unit you banish back into the same showdown?

No, you cannot play the unit back to the same battlefield in this situation.

According to the card text and established rulings, Arcane Shift requires you to target both a friendly unit AND an enemy unit at a battlefield to be played. If you have moved your unit to an empty battlefield, there are no enemy units present at that location for you to target.

As stated in FAQ #8655:

"Arcane Shift requires targeting both a friendly unit AND an enemy unit at a battlefield - you cannot play it without valid targets for both. If you have moved your unit to an empty battlefield, there are no enemy units present at that location for you to target."

Furthermore, even if you were in a situation where you could legally play the spell (e.g., if there were an enemy unit present), you generally cannot replay a unit to a battlefield you do not control. When you move to an empty battlefield, you are the attacker, and you do not control that battlefield until the showdown is resolved and you have successfully conquered it.

As stated in FAQ #8609:

"As the attacker, you do not control that battlefield—the defender does, even while contested. You can only play the unit to your base or to a battlefield you control."

Because you do not control the empty battlefield you are attempting to conquer, you cannot play the unit back to that location.