The problem with that is that it means that consciousness is not bound to your body. Two people can share the same consciousness and from that point on it is very easy to imagine that you are sharing the consciousness of your parents and their grandparents and so on until every organism on earth shares the same consciousness. This means consciousness has to be a property of the universe. Making a clone doesn't grant you immortality because you already are immortal.

The opposite scenario is that there are now two consciousnesses. If the original dies so does its consciousness and you failed to achieve immortality. Your clone is just your offspring with an identical body and set of memories.

So which is it? I personally picked it based on what I want to believe in, because even if it is a lie it helps me shape the world into one in which I want to life in, rather than one that I do not like and would prefer to get rid of.