Good examples of this would be in Troy (Ajax vs Hector) and (Hector vs Achilles cousin) and plenty of other movies.
In the battle of Kulikovo, between the forces of the Grand Duchy of Muscovy vs. a Mongolian army, it is purported that both sides sent out a champion in single combat, and that both killed each other in the first pass, although according to the Russians the Muscovite champion did not fall from the horse while his opponent did. As I do not know of any Mongol sources on this battle, this claim should be taken with a grain of salt as there is obvious incentive for Russian sources to glorify their countrymen.
Source: Martin, Medieval Russia.
The champion-style, heroic warfare was common in pre-Classical Greece. Showing courage and strength in battle was the best way to distinguish oneself, and it was way for the leaders of warrior bands to legitimise their leadership:
The chieftains of war bands investing Troy in the Iliad espoused a code of honor, whereby each strove to prove himself the best (aristos) in a display of martial excellence (arete). Valor in battle validated leadership of the group, and arete directly related to a warrior's Risikobereitschaft [willingness to take risks]. In this society respect among one's peers mattered, as did the glory (kleos) of an individual passed to future generations. (Wheeler, 122)
Of course the most honour and glory could be gained from defeating the enemy champion, so they would seek each other out even in the midst of battle. This could even lead to the battle dying down a bit as others would stop fighting to watch the (informal) duel. So the answer to your question is: yes, there was a period where this did happen.
This practice ended quite soon though. With the development of the hoplite phalanx, there no longer was any room for individual bravery. The requirement to stick with the group and maintain formation precluded any individual actions. Furthermore, battles started to involve more men and became more complex affairs. Gradually the leader of an army became more like Odysseus - intelligent, clever, tricky - and less like the heroic warrior Achilles. This was a process of several centuries, so you can still something of Achilles in Greek/Hellenic generals of the 5th and 4th century (Alexander the Great being a prime example) and even some Roman generals of the 3rd century, and they still would get involved in the fighting (usually leading cavalry), but you wouldn't see them seeking out the enemy general and dueling with him.
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