If I use Relentless Pursuit on my opponent's turn, can I move the unit back to base after conquering even though it's not my turn?

Yes, you can move your unit back to your base after conquering, even if it is currently your opponent's turn.

Reasoning

The "When I conquer, you may move me to my base" ability is a triggered ability granted to the unit by the resolution of Relentless Pursuit. Triggered abilities function independently of whose turn it is. As long as the unit meets the condition (successfully conquering a battlefield), the ability will trigger, allowing you to resolve it and move the unit to your base.

Sequence of Events

1. Playing Relentless Pursuit: You play Relentless Pursuit (which has the [Action] keyword, allowing it to be played during a showdown on an opponent's turn).
2. Resolution: The spell resolves, moving your unit to a battlefield and granting it the triggered ability: "When I conquer, you may move me to my base."
3. Conquest: Your unit conquers the battlefield.
4. Trigger: The granted ability triggers. Even though it is your opponent's turn, you may choose to place this triggered ability on the chain.
5. Resolution: When the ability resolves, you perform the move to your base.

Key Clarifications

• Trigger Timing: As noted in RiftJudge FAQ #9079, the move to your base is a triggered ability that occurs upon conquest. Triggered abilities are not restricted by whose turn it is.
• Ready Status: According to RiftJudge FAQ #9636 and RiftJudge FAQ #9079, this movement is an effect of the spell, not a "Standard Move." It does not consume the unit's "ready" status, nor does it require the unit to be exhausted to perform the move.

Note: I could not fully verify the interaction regarding previously scored battlefields against the retrieved rules.