The problem is rather the interface between rail and road. Now simply outmoded for the current level of use for both. I think that moving the road is likely to be a lower cost solution.
A by-pass using approach roads and ramps to a tunnel under the station have been suggested in the past.
Some of the issues include:
1. Intrusion into Dedham Vale AONB. Map here:
http://www.dedhamvalestourvalley.org/assets/Maps/Project-map-high-res.pdf
2. Dealing with water table and drainage. But as
@306024 notes above the Dutch seem to be able to manage ok!
3. Funding. Using the Werrington grade separation project (which does have at least some similarities) as a yardstick then cost could be ballpark £200million.