I'm after X.

[Tedwonny]

I can often hear this phrase:

I'm after something

From the context, I can infer that the speaker is pursuing/looking for something, right?

Are these sentences ok then?

1) Oh, what a cool car. That's exactly what I'm after.

2) I'm after this book. [looking for this book]

Any typical examples? Thanks so much

[BobSmith]

2) I'm after this book. [looking for this book]

[not a teacher]

Both sound fine to me. I think 2) works best if context was already establish, say, you are pointing to a book in a catalog.

"After", to me, can suggest a slightly higher sense of urgency, then say, "want".

That's the car I'm after! (nothing else will do)
vs.
That's the car I'm wanting. (fits perfect, but something else might too)

Also, "after" can have a slightly negative connotation, as in:

1: My Mom married a man half her age.
2: Oh yeah, what do think he's after?

[JMurray]

I agree with BobSmith and here are some more examples.
"I'm after the kind of experience that will get me a career in catering".
"What book can you recommend? I'm after something life-changing".
"She was after the truth about her father, was he really a Soviet spy?".
"The bright red chairs finally gave the room the effect she was after".

not a teacher