Do buffs stay after Zhonya's on a unit that is buffed?

Yes, a unit retains all of its buffs when it is saved from death by Zhonya's Hourglass.

According to RiftJudge FAQ #4022: "When Zhonya's Hourglass saves a unit from death, the unit keeps its buffs because the unit never actually dies."

Because the unit is recalled to its controller's base rather than moving to the trash (a non-board zone), it does not "leave play." Per Rule 705, a unit only removes its buffs when it leaves play. Additionally, while Zhonya's clears damage from the unit, it does not remove other status modifications like buffs or debuffs.