For Relentless Pursuit does "when I conquer you may move me to base" effect remain for the rest of the turn?

Yes, the effect granted by Relentless Pursuit—"When I conquer, you may move me to my base"—remains active on that unit for the rest of the turn.

As specified on the card, the effect explicitly states: "This turn, that unit has 'When I conquer, you may move me to my base.'"

Because it is a continuous effect with a duration of "this turn," it will apply every time that unit successfully conquers a battlefield for the remainder of the turn (subject to the standard restriction that you can only score each battlefield once per turn, as per RiftJudge FAQ #7610 and RiftJudge FAQ #9258).