So after Doctor number 3 (Hugh Dennis) finally recovers from his partial amnesia and gets to travel through time and space again with his new companion Kathy... the unthinkable happens.
One of his sworn enemies (the Daleks/Cybermen/etc.) engineer a plot that will strike the Doctor's heart. The aim is to break his will before breaking his body. Kathy gets hit in the crossfire and is dying. To save her life, the Doctor siphons off some of his own regenerative power into her. It works, but the Doctor's own regeneration process is triggered. In a flash of blinding light he regenerates into the 4th doctor, played by Rory Kinnear!
The characteristics of his Doctor may be read in more detail here, as this incarnation is mostly Bobby's idea. I had considered Kinnear in the role, but Bobby's imagined portrayal sounds like a lot of fun.
Essentially a much more reserved and timid Doctor, a stickler for etiquette and manners, but not without a more fun-loving side (which he keeps well hidden, but is sometimes revealed thorugh his colourful odd socks). Surprisingly good with kids, I'd see his two main companions being (as Bobby suggested) an art student and a child. He'd pick up Carla, the 25 year old art student, in 1970s Milan, while visiting a museum (perhaps in an attempt to correct the date on one of the more important artworks to avoid some sort of mundane paradox) and 12 year old Ricky in 2021 London. Throughout their adventures, both companions would attempt to get their Doctor to relax a bit more, sometimes with disastrous results.