I also second Sungkyunkwan Scandal, because it has the best bromance ever between the playboy and the rebel *smirk* I still think the former is actually gay but hides it behind his playboy facade. Ahhh, they were my favorite characters in that drama. Also, it isn't a boring historical drama, it has a somewhat modern feel to it, I think you'd like it :)
'I remember you' also has touching bromance!
I can recommend 'It's okay, that's love'. Even though I am not sure if you can say it's actually bromance, it's definitely a very deep friendship, akin to two brothers. It made me cry at parts.