It sounds like you aren't very happy at your current job, so by all means look for a new job. That said I wouldn't like the stress of starting a new job immedia

It sounds like you aren't very happy at your current job, so by all means look for a new job. That said I wouldn't like the stress of starting a new job immediately after recovering from surgery. Why not request leave and after you come back start looking for a new job? As for notifying them that you have died, maybe you can ask a close friend or family member to do it? (Edited to add: in the case that recovery takes longer than expected you definitely want to be employed, so quitting your job and starting a new one is maybe not advisable.)