As seen in Journey's End, a Meta Crisis Doctor was created by Donna Noble after she touched the Doctors handy hand in the container. The Meta Crisis Doctor is half Time Lord, half human as he only has one heart. But he stills has the knowledge of a Time Lord. We saw Rose Tyler going to live a normal life with the Meta Crisis Doctor. They would both age together and grow old together. But where are they now and what are they up to now? A spin off show would be ideal to check up on them.
What Would The Spin Off Show Actually Involve?
It would be similar to the mini Spin Off about Amy and Rory which is on the DVD Box Set. It was called Pond Life. Basically just showed us viewers what Amy and Rory got up to without the Doctor. If you haven't seen Pond Life, you should. So yeah, the Meta Crisis and Rose would be living together. They've moved out from home (where Jackie and Pete are) and have their own house somewhere in the countryside and just living together.
But this is where it gets exciting. They wouldn't live an ordinary life, they'd be battling enemies just like old times. The Doctor would be able to construct together his own Sonic Screwdriver. The Doctor would be able to put together a Sonic Device for Rose too. So off they go. Only difference is that there is no TARDIS maybe a car. They would have to track down the enemies down just by the events that are happening around them. For example things/ people going missing and strange activity happening where they live.
This is a great example to see the return of many Doctor Who enemies like, Daleks, Cybermen and Sontarans. Even bring back some of the enemies. monsters we haven't seen in a while like the Autons, Judoon and maybe the Vashta Nerada.
This also could fill in the time whilst Doctor Who isn't broadcasting a current Series. Then Whovians would have things to watch whilst waiting for the new Series to begin.
This Sounds Similar To The Sarah Jane Adventures...
You're absolutely right, it does sound like the Sarah Jane Adventures. Sonic device, car, battling enemies without the TARDIS. But the Doctor and Rose won't have a Super Computer like what Sarah Jane has. The Doctor would use his Time Lord knowledge to help him out. Unlike Sarah Jane the Doctor still has all that knowledge flying round his head. Sarah Jane gets her information from Mr Smith. So yes, it's similar but something different.
Your looking at something about 25-30 minutes per episode. This then gets the stories spaced out and wouldn't be crammed together in 10 minutes. You're also looking at about 8/9 episodes per series too. The episodes would air weekly.