What happens if I attack a BF, my opponent uses Zenith Blade on my unit, and I stun his unit, so we both are stunned?

To answer your question accurately, let's break down the timing and combat rules. Based on your description, you are the Attacker (you moved into the battlefield first, establishing the Contested status).

The Sequence of Events:

1. Your Attack: You move your unit to an empty battlefield. You are the Attacker, and the battlefield is now Contested.
2. Zenith Blade: Your opponent plays Zenith Blade during the showdown. They target your unit to stun it and move their unit to your battlefield.
3. Your Response: You stun the opponent's unit (e.g., using a [Reaction] or [Action] card available to you).
4. Current State: Both units are now present at the battlefield and both are Stunned.

Combat Resolution:

According to Rule 423.1.b and Rule 460, a stunned unit does not contribute its might to combat damage. Since both units are stunned, neither unit will deal damage during the Combat Damage Step.

• Combat Damage Step: Because both units have 0 damage output, no units are killed.
• Combat Cleanup: Your unit is the Attacker and the opponent's unit is the Defender. Rule 461.1.a.2 states that at the end of combat, "Recall Attackers present at the Battlefield if Defenders are still present."
• Result: Because your unit (the Attacker) is still at the battlefield and the opponent's unit (the Defender) is also still there (since it was not killed), your unit must be recalled back to your base.

Who Conquers?

According to Rule 461.5 and RiftJudge FAQ #1050/7095:

1. Your unit is recalled to your base.
2. The opponent's unit remains on the battlefield alone.
3. Because the opponent's unit is the only one remaining after combat cleanup, they establish control and successfully Conquer the battlefield.

Summary: Even though both units were stunned and survived, because your unit was the Attacker and the defender remained, you are recalled, and your opponent conquers the battlefield.