I started this book two months ago.
I've started this book two months ago.
You began reading the book in the past, so you want to use the simple past tense. I've = I have, which is the present perfect tense. The present perfect is used for an action that began in the past and came to completion in the present.
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You began reading the book in the past, so you want to use the simple past tense.
I've = I have, which is the present perfect tense. The present perfect is used for an action that began in the past and came to completion in the present.