I agree that you need to keep it Simple Stupid, but here you can use most of the Scrum infrastructure.
I had several people working in this way both on projects and on maintenance work.
Product Owner / Backlog - There is still an owner who is responsible for determining the value of the business and setting priorities, right? There must be a backlog. If he is part of a larger Scrum enterprise, he will probably need to feed part of a larger Backlog product.
Team Scrum - yep, her team of 1 or 2 people. So its really a SELF organization ... but that's fine! Daily fight? Yes, between two people, or if it’s just one person at times, a good time to solve problems and problems, think about what obstacles you need to come up to Scrum Scrum or the owner of the product.
Sprint is still a good idea, especially if it is part of a larger Scrum enterprise that works in the sprint, but even without it. A good chance to catch up with the PO, the demo that you got, activate yourself, retrospectively and see what you can do better, plan your next sprint. Please note that if you work outside the Scrum / Scrum of Scrum facility, the sprint may benefit from being shorter than usual, since the scope is probably smaller and the planning overhead is lower. but it depends on the situation.
Retrospective - yes, it can be done alone. I think killer programmers should look back at their work / progress and take action on what keeps them going. Even keep a chart in your workspace to help you make progress.
Taskbar / Burndown - Yes, you need it. You can have them in your workspace on the wall, they can be small, but they really help, even if you are one person. Why does GTD (Getting Things Done) help one person and TB / BDC? If this person does the work on the project, then Sprint Burndown and Release Burndown give great value. If he carries out operational / technical work, this is another way to check it or not, and apply appropriate measures accordingly.
Scrum Master - a person must be his own master of the battle.
Coach - if the organization had a coach helping teams / SMs / POs, then he also had to help this fight ...
To summarize, it is clear that the values and principles that underlie Scrum / Agile are also applicable to teams of 1-2 people. It is also clear that much of Scrum can also be applied.
Questions are what people think.
If the management, developer, software are all on board and believe that the values / principles make sense and strive for improvement, it will work. If they do not, then first get to the point that general thinking makes sense, and then figure out a separate team ...