A lot of recumbents, including the Cruzbike Sofrider pictured below, allow you to ride in the same kind of seated position you do in an office chair, and the ones that have high bars (instead of having the handle bars under the seat) allow you to mount rear view mirrors so that you needn't turn your neck to look backwards.
Recumbents are lower to the ground than upright bikes, which makes them a fast efficient alternative to "comfort bikes", though it also makes them harder to use than a road bike for fast city riding in dense traffic since you can't stand up to see over cars.
which are the best for:
Recumbents are great for riding in flat areas, though not as good for hilly areas -- they're not quite as stable at low speeds since you can't as easily reposition your hips, and you can't stand up to torque when you run out of gears.
mountain bike, european city bike, touring bicycle ,recumbent bicycle
Mountain bikes can be ridden with back and neck straight but they're designed for people who ride with a slight forward tilt, and they're not efficient on roads.
Many European city bikes are similar to US Comfort bikes, and others like Dutch cargo bikes are designed with carrying capacity in mind. They're good for riding wearing everyday clothes at a pace that isn't going to get you all sweaty, but are no good for recreational riding with people on faster bikes.
Touring bikes cannot as they have similar geometry to road/cross bikes.
As explained above, recumbents come in vastly different geometries, but many do allow the kind of riding posture you describe.