And as a follow up on this along the same theme....if the King or leader is killed would that be the end of the battle?

Hi! I've got an example in the middle ages, not the ancient world though.

I recently read an article by Dominique Barthelemy about Chivalric warfare between the tenth and twelth century (You can find the full source at the end of my post).

He writes:

"In practice however, such campaigns were limited to pillaging, and when Otto's vassal army came up against its counterpart in Hugh Capet's men, the result was not a battle, but one-on-one combat."

One page later again, he writes:

"Similarly, when faced with the vassal army of Otto II in 978, Hugh Capet did not join battle against him; he simply elected and paid for a champion to answer Otto's provocations in one-on-one combat with a German and to kill him with a battle-axe."

As a source he uses Richer of Reims Historiae.

I hope this answers your question at least partially, even if it's not about the ancient world.

Source: Barthélemy Dominique: Chivalric One-Upmanship in France, ca. 1100, in: European Transformations. The Long Twelth Century, Notre Dame Press 2012

(This is my first answer post on this subreddit, I hope it's enough to hold up to the high standards on this sub)

Follow up question: was there a specific culture that allowed for one on one battle to decide a war? The main reason I ask is Netflix's Marco Polo. In episode 2, Kublai Khan defeats his brother (maybe half brother or cousin) in a one on one battle in front of their respective troops and unites the Monguls in one fell swoop. Obviously, I'm not taking the show and base historical value, but rather curious as to the frequency of these situations.