If a battlefield says, you may choose a unit, and send it to base enemy or friendly, is that choosing and therefore deflect cost?

Whether an effect that says "you may choose a unit" triggers Deflect depends on whether that effect is targeting the unit.

According to Rule 355.7 (as implied by the provided context and clarified in Rule 355.10), when a card chooses a specific game object to affect, it is considered targeting unless otherwise indicated. Because Deflect (Rule 809.1.c) specifically applies to spells and abilities that "choose" a permanent, any effect that requires you to select a specific unit as a target will trigger the Deflect cost.

Does your specific effect trigger Deflect?

1. If the effect targets: If the ability requires you to select a specific unit on the board to move it, you are "choosing" that unit. Under Rule 809.1.d, this imposes a Mandatory Additional Cost if that unit has Deflect. You must pay the Deflect cost to finalize the ability on the chain.
2. If the effect does not target: Some effects do not target because they do not "choose" a unit when the ability is placed on the chain. For example, if an effect instructs players to make a choice during resolution (like King's Edict, see FAQ #8987), or if it affects all units at a location without selecting specific ones (see FAQ #5522), it does not trigger Deflect.

Summary of the Interaction

• Targeting: If you must select a specific unit to move, you are targeting. If that unit has Deflect, you must pay the Deflect cost as an additional cost to place the ability on the chain (Rule 356.2.a.2).
• "May" vs. Mandatory: The fact that the ability says "you may" does not change whether it targets. It only means that you have the option to place the ability on the chain. If you choose to use the ability, you must pay all costs—including the Deflect cost—to finalize it. If you cannot pay the Deflect cost, you cannot choose that unit as a target (FAQ #8660).

Conclusion: If the ability requires you to select a specific unit to move, it is targeting that unit. Therefore, if that unit has Deflect, you must pay the Deflect cost to use the ability.

This is my interpretation based on the provided rules regarding targeting (Rule 355.10) and Deflect (Rule 809.1). No official FAQ exists for this exact phrasing, but the interaction follows the established mechanics for targeting and mandatory additional costs.