Multiverse

What is the likelihood of discovering alternate realities/ parallel dimensions?

Is there a possibility of viewing one at any point within my lifetime?

What about tesseract universes?

Scientists say that we can only see as far as the universe is old. The universe is estimated to be 14 billion years old. If that's true we can only see 14 billion light years away, so there is a horizon 14 billion light years away that we can't see beyond. Some even conjecture that ours might not be the only universe, and some theorize there may have been other universes created the same way ours was. And since we can't see beyond the horizon, no one knows what's on the other side.

So I don't know, but I'll leave it for bigger minds.

Within our lifetimes? Highly unlikely, but we can hope

It's not possible, simply by definition: Our universe encompasses everything that can be observed (literally or theoretically) and interact. If we can observe it or interact with it, it's not a parallel universe - it's part of our own.

The interesting thing about the multiverse is not so much that we could potentially see it, but that we may be interacting with it at this very moment. Consider this for minute:
Everything is in balance in the universe, from forces down to quantum particles. Push vs. pull, even electrons vs. positrons. Now consider that even you may have a anti-self, a being created to keep everything in balance. So, with this in mind, everything you do, everything you dedice, choose, agree, or disagree with, alternates your form, potentially creating a division. Where then is the balance? The multiverse postulates that an alternate reality may be created each time one does this, in order to maintain balance. Now there is something to keep you thinking for a while.