On debian 9/10 ibus works for the Korean IME (both Hangeul and Hanja input).
Windows (10 at least) has native support for these IMEs; setting it up on Windows was much easier than on Linux (which doesn't even have a standard IME). Windows also comes with basic CJK fonts, which different distros may or may not have. On Debian I have to install noto-cjk or Adobe's source han fonts.
I originally tried fcitx but had to switch to ibus. Definitely not a great experience.
There isn't a single IME setup experience on Linux.
In my experience, Debian is worse than Fedora, Ubuntu, and openSUSE. Fedora is better than Windows 10 when it comes to IMEs: Fedora installs the IMEs by default. Ubuntu, like Windows 10, installs IMEs when you request the addition of an IME-requiring language. OpenSUSE gives you an IME for the language you use at install time if you install openSUSE using an IME-requiring language.
I haven't tried Debian 10, but when I installed Debian 9 _in Japanese_ with a _Japanese keyboard layout_ chosen, the installer didn't bother to set up a Japanese IME!
Fedora comes with an OK set of Noto CJK fonts by default. Ubuntu comes with a minimal set of Noto CJK fonts by default, but when enabling Chinese, Japanese, or Korean, Ubuntu drops more language-appropriate fonts on the system, like Windows 10.
Windows (10 at least) has native support for these IMEs; setting it up on Windows was much easier than on Linux (which doesn't even have a standard IME). Windows also comes with basic CJK fonts, which different distros may or may not have. On Debian I have to install noto-cjk or Adobe's source han fonts.
I originally tried fcitx but had to switch to ibus. Definitely not a great experience.