K-indie! I listen to some that are quite indie- I think anyway and most of the bands mentioned above are always in my recommendations so I'm going to put my favourites here :)

Urban Zakapa, 10 cm and Standing Egg have been mentioned

Here's a good one by Kwon Soon Kwan: https://www.youtube.com/watch?v=xZy-XalONV

By Sung Si Kyung: https://www.youtube.com/watch?v=5hINYNZslP0

Jang Kiha & The Faces are my absolute guilty pleasure! - Their songs are about anything and everything, make sure to check them out and when you do make sure to watch their music videos too! I shall put the link to some down here so its easier ;) *Bonus Jang Kiha plays a hilarious quirky character that makes up some ridiculous songs like these in Potato Star 2013QR3 - its quite a long sitcom so maybe just see some of its funny clips to see his character and if you like it watch all of it!

I Almost Had it: https://www.youtube.com/watch?v=6Mn26Pp_eHo

Kieuk: https://www.youtube.com/watch?v=fih0WB86zCk

and Cheap Coffee is also one you shouldn't miss! 
More rock than indie but...

Royal Pirates: One of my all-time favourites, it's a shame how underrated they are. They even have english releases as the members are all fluent english speakers.

Lunafly: Another underrated group with unique voices



recently i'm listening to OOHYO, but you must hear Busker Busker (or Jang Beom Jun) and NELL, they never fail