The Haber process is already operating close to maximum possible efficiency, so there are no big gains to be had.
The haber process doesn't require that the hydrogen comes from natural gas, and if you use other hydrogen sources and clean energy doesn't release CO2. From what I understand the popularity of natural gas simply comes from natural gas being one of the cheapest sources of energy. If you have cheap electricty you can just run a fuel-cell backwards, or electrolyse water. Electricity just happens to be more expensive than natural gas. Other processes will hit the same problem, since the overall energy requirement is the same. Whatever process we find needs the energy from somewhere.