Your baby is a hobby!
Your beautiful child is a commitment. At 15 months, you've been through the infant stage, one of the tougher ones. Your child is around the point where he'll/she'll be walking.
This is the fun part!
You're almost at the point where you can take your baby to the park. He'll want to take little walks outside, or get pushed in a buggy. What on earth could be more exciting than having a little person who trusts you and wants to be with you and spend time with you?
Your child will grow. Pretty soon, he'll start finding his own hobbies. Then he'll decide what he wants to do when he grows up. Then he'll be gone. You will have, in theory, all the time in the world. And, unless I miss my guess, you'll hate it. You will miss this, believe it or not. This stage only comes once in your child's lifetime. Savor it. Make it a hobby.
As for other things that you can do when you have a spare minute ('cause yes, you still get to keep a little of your sovereignty ;P) I suggest getting a kindle. I love my kindle. You can take it with you to lunch and read while eating, between your and your child's bites. You can have it in your lap in a chair while watching your child play in the living room. It's a great tool.
While learning a musical instrument has been suggested, given your current situation and the neediness of the child, this seems like it could present a bit of a challenge. However, listening to music is an amazing thing. And by music, I don't mean just pop. Branch out! Try some classical music, or even some jazz! Do a little research and find out what to listen for in whatever category you decide you listen to. Become an expert! Your music player will be your best friend.
God bless you and your child!