Technically no.The Fighter class − in order to be "nonmagical" − needs to become a prestige class that only has levels 1 to 12. There is no level 13 Fighter. Because level 13 is inherently magical.
The Tier 3 and Tier 4 fighter isn't magical. It's just not simple. The desciptions and depictions of master swordsmen and archers ad details that are not simple.
The Captain America Winter Soldier Elevator Fight scene starts with Cap
- Defending a grapple check with 10 guys using the Help action
- Beating the STR Saving throw on a
high techmagic handcuff
- Making 8 unarmed strike. Each causing a knock down.
- Doing 2 again
- Dropping 3 Tier 2 warriors
That's more that the +1 more to Proficency and extra use of Indomitable you get at level 13. The Fighter can't level up and stay both simple and mundane. That's one of the core problems. It's easier to make a Mythic Warrior stay simple that a Mundane Warrior.