I tend to agree with sbytheway. The first people to become 'amortal' may be super-rich, but I doubt normal western people would sit around for long not trying to get such a valuable 'commodity'. In western countries where we mostly have healthcare provided, i do not think it would be a stretch for people to want to make amortality provided for as well. It might not take long for people to start calling life (long life) a right of the citizen in such countries. That said, there would be people in other countries which do not have the means to provide this. There, only the rich might receive treatments (provided in western hospitals, perhaps), and when the poor see their loved-ones dying while westerners (not to mention the ruling elite of their own country) avoid this, there will be resentment. Some off this resentment may boil over into violence--and even if it doesnt, the newly Amortal will fear it more than ever.
I don't think amortality is a bad thing. But from the get-go, it must have universality as it's end goal, or else the two diverging groups in humanity might each inflict a different kind of dystopia on one another.
Healthcare is a universal right (for example, in Italy any medical treatment is free for the poors and have a symbolic cost for the middle class and the rich) because not everyone get seriously ill at the same time: through taxes the State provides healthcare, that means you don't have to pay 100'000+ € for a heart surgery.
But if everyone needed during the same year a heart surgery, the whole country's GDP wouldn't make up for the cost of providing that service to 60 mln people.
Economically, universal healthcare is an universal healthcare insurance: you have a certain probability of getting ill during your life, so the State provide you with the coverage. The point is that the probability is low enough to allow it with just a 15% tax rate (for healthcare alone).
Let's say that providing body-enhancing surgeries would cost 10 mln € for the whole life of a person (300 years, since improved?). It would mean, taking Italy as an example, that you have to spend 10 mln x 60 mln = 600'000 billions €, or 400 times the GDP (and 60-80 times the wealth, both financial and physical, of the population). You can understand it would be impossible to provide such a technology to every person, even making it a universal right.
If the cost is 1 mln €, you would need 40 times the GDP and 6-8 times the wealth.
If it's 100k €, you would need 4 times the GDP and 0,6-0,8 times the wealth. And, over a 300 years life span, it would be possible to provide it as a universal right, making the whole population "immortal" in 30-40 years.
So: really much of the conclusion depends on the cost of such a technology. Even in a non-capitalistic society or in a capitalistic society with a state trying to provide it for free.