There are a lot of threads out there about how to "improve" the fighter...assuming, of course, that you are of the opinion that the fighter class needs to be improved at all.
One problem I've often seen with "the fighter sux" or "the fighter needs more whatever" is that the real balance problem is on the other side of the fulcrum: the caster having too much or being too vastly overpowered. Cutting down casters by a few orders of magnitude would do a lot more to make the game balanced and playable than turning fighters into wuxia/anime/superheroes.
I think the best way to add complexity and flavor to the Fighter class would be to do it with his catalog of weapons and armor. After all, these two things are what have set the Fighter class apart from all the rest in all editions of the game: the Fighter is the guy who can use all the weapons and wear all the armor. So why not focus on that, and make that dynamic more important?
It would be an interesting dynamic, if it were, indeed,
dynamic. That is, if the fighter were likely to use different armor and weapon mixes for different missions, and use them in different ways according to his style and talents. You could have fighters that were both differentiated and versatile, that way. Of course, you'd still have overpowered casters to balance them against.
So what if the fighter could choose a type of weapon (say, Axes) and a type of armor (such as Light Shields) to "specialize" in, he gains a special technique that only he knows...like maybe being able to use the blade of an axe as a rudimentary shield when fighting defensively. And, it grants him extra maneuvers, in combat that only an Axe Specialist would know: stuff like spinning leg sweeps, bonuses against Plant creatures, vorpal blows, and so forth.
So, if the big deal of the fighter is his ability to use any weapons or armor, the way to build on that to make him more dynamic and interesting is to take away most of those in return for being uniquely better at one?
That's prettymuch what 3e did. It took the universe of everything a fighter-type might do, cut it up into dozens of feats, and gave the fighter 11 of them.
A more interesting approach might be to have 'styles' that are much broader than using a single weapon or weapon-and-shield combination. Each style would have a variety of techniques - maneuvers or exploits or whatever semi-mythical vaporware 5e is going to use to somehow balance the fighter with the Vancian casters we've already seen - some usable with certain sorts of weapons.
The best example might be something like a Samurai. They used swords, pole-arms, and bows, a single two-handed weapon, or a weapon in each hand, relatively heavy armor or none at all, mounted or afoot, in pitched battles or formal duels or in self-defense against surprise attacks. A Samurai wielding a naginata would not use the same techniques as one fighting a duel with katanas or one using a bow from horseback.
By choosing the weapons and armor he employs in a given circumstance, such a fighter would be able to tailor his abilities, much like casters do by memorizing particular spells. You could even get away from the fighter's excessive heavy-armor dependence and open up more skills and exploration tasks that he could be good at, particularly when opting for lighter armor. A high level fighter with a bag of holding could carry a wider variety of armors and weapons, making him more adaptable, and really interesting/powerful items, like the classic Rod of Lordly Might that changes into several different weapons could give him even greater flexibility.