I agree it should be kept Simple Stupid, but most of the Scrum framework can be used here.
I had several people working in this fashion both on projects as well as on maintenance/operational work.
Product Owner/Backlog - There's still an owner thats in charge of defining the business value and prioritizing, right? The backlog should still be there.
If he's part of a bigger Scrum enterprise then he probably needs to feed on part of a bigger Product Backlog.
Scrum Team - yep, its either a 1 or 2-person team. So its really SELF-organization... but thats ok! Daily scrum? yes between the 2 persons, or if its just that 1 person at times, good time to go over tasks and problems, think about what impediments need to be surfaced to the Scrum of Scrum or to the Product Owner.
Sprint - Still a good idea, especially if part of a larger Scrum enterprise thats working in sprints, but even without it. Good chance to catch up with PO, demo what you got, energize yourself, retrospect and see what you can do better, plan for the next sprint.
Note that in case of working outside of a Scrum enterprise / Scrum of Scrum, sprints can benefit from being shorter than usual since the scope is probably smaller and the planning overhead lower. but it depends on the situation.
Retrospective - yes, it can be held alone. I think killer programmers need to retrospect on their own work/progress and take action on things that hold them back. Even keep a chart in your workspace to help you with making progress.
Task Board / Burndown - Yes, you need those. You can have them in your work space on the wall, they can be small, but they really help even if you're one person.
Why can GTD (Getting Things Done) help a single person and a TB/BDC not? If that person is doing project work then a Sprint Burndown and Release Burndown provide a lot of value. If he's doing operational/maintenance work its still a way to verify he's on track or not, and apply relevant measures accordingly.
Scrum Master - the person should be his own scrum master.
Coach - if the organization had a coach helping the teams/SMs/POs, then he should also help this scrum cell...
To sum up - its clear to me that the values and principles underlying Scrum/Agile apply for 1-2 person teams as well.
Its also clear that most of Scrum can be applied as well.
The questions is what the individuals involved think.
If the management, the developer, the PO are all on board and believe that the values/principles make sense, and strive to improve, it will work.
If they don't, then first get to the point where the overall thinking makes sense, then deal with the individual team...