I actually thought about making this recently (My version didn't have completely unique units, some civs shared a UU with another, others shared names).
In addition to unique units, I was also thinking about Unique Wonders (one for each civ)
It would add a lot of flavor to the game, if done correctly.
For Carthage, how about a slight modification, plus a UU.
Change Numidian Mercenaries to Libyan ones. Numidians were famous for Light Cavalry and Light Skirmishing infantry. Libyans were the Heavy Spearmen preferred in the Punic Wars (In case you don't know, the art files say Libyan Mercenary as well). The other UU would be Numidian Cavalry. This light, Javelin-throwing Cavalry was probably the best light cavalry in the western Meditaranian.
Byzantine -Cataphract, obviously
Germany- U-Boat
Greece -Trireme
Viking - Longboat
Iroquois - Either an Archer or a Gunpowder unit. Iroquois boys were trained to use bows and arrows from an early age, but it wasn't until Gunpowder that the Iroquois grew to tremendous strength.
Spain - Tercio. A combination of Pikemen and Gunpowder units was used very effectively by the Spanish (probably a Gunpower replacement with an extra defense)
Persia - Horse Archer. Light Cavalry used effectively by all Middle-Eastern Empires from Assyria onwards (also used by Steppe civs, like the Huns)
Rome - Although the Pretorian Guard has been mentioned several times, the Guard really wasn't practically used. An emperor had to pay the Pretorian Guard, or he would be assasinated. They were nowhere near as important as Legionaries. They wouldn't really fit as Spearmen because they came after the Bronze Age. What about Axillia, making a Swordsman that requires no resources, for those unlucky Romans without iron.
If you can't think of unique units for every civ, I also gave Babylon Horse Archers, and Sumer War Chariots.