I recommend Severance Hospital located in Sinchon-dong, Seodaemun-gu.

You probably already know that this is a very famous hospital. Many medical students trained at Yonsei University are known to come here for internships and continue their careers as doctors here. Severance Hospital is said to be filled with highly skilled doctors, and it wouldn't be an exaggeration to say so. The department I visited was plastic surgery. The doctor I met was very kind, and I received good medical care and scheduled my surgery date smoothly!