I had the same idea about the phone years ago. I don't know why we can't just take our iPhone or (more likely) Android, plug it into a docking station and use that at work as a desktop, pick it up and use it as a smartphone, or dock it to something at home.
I have a lumia 950 with continuum, and it does this even without the dock. You wirelessly project to any w10 device, and it's like plugging in a different compute engine. You get a desktop, the start menu is your phone's home screen, all apps adapt to the bigger screen, and you can multitask. You use the keyboard and mouse of the PC.
It has two major downsides right now. There's no win32 support, so there's not much to run on the desktop, which MS is going to address by bringing x86 emulation to arm. Secondly, the hardware isn't quite powerful enough, which limits the max. resolution and the ability to multitask, which the new qualcomm 835 based phones with 4+ gb of ram should address.
It is impressive, but I haven't found a use for it. After all, when you have a laptop, why use it to run apps from your phone?
As mobile chips continue to improve, emulating x86 becomes a viable way of tapping into the huge Windows ecosystem of applications when not running on battery power.
I think this is Microsoft's last shot of becoming relevant in the mobile space.
Bring your own screen, keyboard, and power supply. Continuum is a weird beast, I'm not exactly sure who would get something out of it even if app support was good.
If the phone hardware was more powerful, and the win32 app support was there, it could replace your PC. You'd have a USB-C dock, with keyboard, screen and mouse, into which you plug your phone. Your phone is your PC, and when you leave you take it with you along with all your files and all your software.
You could combine it with something like a nexdock to also turn it into a laptop. Would be cheaper than getting a phone, and a desktop pc and a laptop, and it would avoid all those nasty issues with keeping devices in sync.
The qualcomm 820 already performed at core m levels, and the 835 is 25% faster, so it should be fine if paired with enough ram and storage.
I recall Nokia demoing something like that with their later Symbian phones. Video out to a TV, bluetooth keyboard and mouse, and you got the phone UI up on the TV and could operate it much like you operated a desktop. And you could still see and interact with the UI on the phone screen.
Android was out at that point, but could barely do video out by blanking the phone screen. Forget about operating it via other means. And i don't think iPhone was doing much better.
One reason why i feel the focus on those platforms reset mobile device progress by a decade.
Speaking to retailers recently, demand is for office apps usable on touch. i.e. if it can possibly done without a keyboard, people will do that. From listening to users, they will moan and complain about touch "productivity" apps... but will choose it.
Personally, I switched entirely to a phone a year ago... though with a bluetooth keyboard.