When you are up to neck in something, you are very busy. Read the piece of conversation given below.
I’m up to my neck in work
Storyline: Aditi has been working at her desk for a long time. So her friend Priya suggests that she should take a break.
Priya: Let’s take a coffee break, shall we?
Aditi: I’d like to, but I can’t.
Priya: Why? You’ve been sitting at that desk since morning. You haven’t even had your lunch. What keeps you so busy?
Aditi: I’m up to my neck in work. I’ve got to finish this assignment today itself. If I don’t, the manager will show me the door.
Priya: I understand, but you can’t sit on that computer forever. You need to take a break every now and then.
To show someone the door is to dismiss them.