Calvin agrees that it's common, Ben says it's wrong. This seems to suggest a regional difference. In informal British English, it's certainly normal and very common to use "be after something" to mean "be looking for something". For example, if you see a friend or family member rummaging through a cupboard or drawer, you'd ask "What are you after?", meaning "What are you trying to find?".

"I'm after a book" is not common. I understood it though because I knew what you meant to say - your intent. You should say though: "I'm looking for a book." I think the meaning of "after" you're thinking about here is in, for example - "I'll look after your son/daughter" - which is a way of saying you'll watch over them and protect them. You'll look after them.

Yes, and your example is correct. 'to be after something' = 'to be looking for an object/person' It's a relatively informal expression and fairly common: "I'm after a new jacket" It can sound aggressive/accusative if you use it with people: "I'm after the guy who stole my wallet!"

Yes, you are correct. We can use "be after" when talking about something we are looking for. Like the example you used: "I'm after a book by Jane Austen called Pride and Prejudice. Do you know which bookstore has it?" (meaning I'm looking for it). Or the common question: "What are you after?" (Meaning what are you looking for or what do you want to accomplish).

'I'm after you,' can have two meanings the one everyone here has mentioned "looking for something" and the other is referring to taking turns in something like a game you go first then its my turn 'I am after you.' Its an orderly progression like in a queue the man in front is before me and I'm after him.