I am not remotely convinced that "transplanting consciousness" is a thing that is even possible

At best we may eventually be able to copy a consciousness, but that isn't the same thing

That would make an interesting story plot. Suppose we've developed the ability to copy a consciousness. It has all your memories, all your feelings, your same sense of "self" or identity. If you die, you experience death, but the copy of your consciousness lives on, as a perfect replacement. Would that be immortality?

I have thought about this quite a lot

I don't think it is immortality. It is just cloning

Any theoretical scheme that could let you exist at the same time as a clone of yourself means the clone is clearly not you. It's a different independent individual that only appears to be you

Altered Carbon, Richard Morgan 2002. There's also a Netflix series.

Where does one find a new body ready for consciousness transplant? Would we grow them in farms like in the Matrix?

I think growing a new body is going to be the easy part

How do we separate a consciousness from one body and put it into another?

What would that even mean?

I don’t want to be too confident on something like this, but I feel like consciousness comes somehow from the material body (and surrounding world) in all its complexity, so transplanting consciousness absent transplant of physical material wouldn’t be possible in theory. This assumes it’s a consequence of the structure of things and not something separate, but I think that’s a reasonable guess.

The way I think of it is that consciousness is a side effect that arises from the complex circuitry of our brains

I also don't want to be too confident, I'm not an expert on this. But I don't think consciousness is tied to any one physical component of our brains, it is something that only happens when the whole system is assembled

This is why I don't think you can move consciousness. You can create a new identical brain, but that create a new consciousness. How do you transplant a side effect?

It would be like saying "we can move the heat that this circuit is generating to this other circuit". Clearly you can't really