Hey, @xabi! It's important to keep in mind that biological immortality is very different from how we often see immortality portrayed in science fiction. Biological immortality just let old individuals not be at a higher risk of dying than at their peak. For a human this would mean that a 100 year old has no higher mortality rate than a 30 year old :)
I think it might be a goose chase to try to achieve either types of immortality for humans, unless we consider recreating an AI of one's brain to be immortality.
Cloning ones body and thaughts is an option.
Can we keep someone's brain alive and working even after the body dies.
Yep, that's true. I think that at one point the body will be holding us back from living for a much longer time, so at this point we need to keep the brain/conscious seperated from it if we are trying to achieve immortality.