Netflix has a short series that carries this idea one step further: Altered Carbon. In that series, Earth has gained alien technology that allows people to store their consciousness in pods and then install those pods in a body.
When a body dies, the consciousness is stored in the pod and can be transferred many times to a new body. The contents of the pod can be backed up, stored in a new pod and so on.
It's a pretty trippy story and I found it well worth the time to watch. I tried watching Black Mirror and found it rather nauseatingly negative. I might come back to it again someday, though.
Yeah that is a pretty common, suggested way of becoming immortal once we are able to store our consciousness as digital data. The idea is that stay in a body (biological/mechanical) as long as that body can sustain you and after it no longer can, simply move on to other body. As simple as that!
I will certainly watch Altered Carbon now as I am really interested in this subject and I hope you complete the Black Mirror series as well because even though it can be very negative, it certainly presents ideas that are VERY possible in the future, so better stay informed and think about it.