In 16.04, fcitx is the default framework for Asian input methods, and it's Mozc (fcitx) which is made available by default when you install Korean.
However, if you didn't select Korean in the installer, you probably need to switch from ibus to fcitx to make the Mozc option visible in the UI. You can do that by opening a terminal window and run the command
im-config -n fcitx
Then, next time you log in, Mozc ought to be available.
Is it a possibility to make this switch via a UI you would like to see?
Thanks for your report.
In 16.04, fcitx is the default framework for Asian input methods, and it's Mozc (fcitx) which is made available by default when you install Korean.
However, if you didn't select Korean in the installer, you probably need to switch from ibus to fcitx to make the Mozc option visible in the UI. You can do that by opening a terminal window and run the command
im-config -n fcitx
Then, next time you log in, Mozc ought to be available.
Is it a possibility to make this switch via a UI you would like to see?