Qarthadastim is most definitely my favorite faction in EB. With some smart expansion, you will get access to every unit type in the game: superb heavy cavalry, hoplites, 2 (!) types of elite spearmen, 2 (!) types of elite swordsmen, elite pikemen, the best light cavalry in the game, and elephants all come right from your home provinces. Expand over toward Crete and you get the best archers around. Reach out a bit further and conquer the Bosphoran Kingdom cities, and you get horse archers. Conquer Italy and you get some great Samnite swordsmen heavy infantry, italian light cavalry, hastati, etc. Lots of powerful ships allow you to keep a strong navy that makes travelling from one side of your empire to the other possible in only a couple turns even after you own 50+ provinces. Pile on top of that possibly the best economic position in the game thanks to the many mine-heavy provinces around you, and you have an incredibly powerful faction.

My favorite thing about the Carthage campaign is you start so spread out, which gives you basically limitless possibilities for how you want to expand. You can go literally anywhere you want once you get your economy running. I'm playing Carthage now and in 167BC, my empire is spread out all over the place. I own all of North Africa, Egypt, Iberia, Italy, Crete, the Bosphoran Kingdom, Ireland, and parts of Greece. Because you get access to so many types of troops, you can easily expand to anywhere and beat any faction since you get hard counters to every unit that could possibly be arrayed against you - ample javelin units for elephants and chariots, archers for mass infantry or light cavalry, and all manner of heavy infantry for whatever purpose suits you.

This is the only campaign I've kept playing for so long, other factions I usually get bored of before 200BC.