I'm surprised Microsoft hasn't just killed it off already. This is their third reboot, with nothing to indicate that this time will be any better than the last ones.
I am...not a fan of Microsoft. In fact, Windows 10 (with built in self serving advertisements) is something I consider really abusive to Windows users. But...
I'm looking at Continuum with a lot of interest. In my younger days, I loved to build computers, and tweak things, and try alternative operating systems, and upgrade my systems part by part, etc.
But these days, I mostly want things to Just Work with as little hassle as possible. Continuum has me interested because I'd love to have just one computing device -- a smartphone -- that can also be easily attached to an external display/keyboard/mouse, and offer me desktop style applications as well.
Store most of my data in the cloud, and I've got a nice, problem free, easy computing experience. That sounds very attractive to me.
I'm not sure if Apple or Google are looking into something similar. Microsoft may be the leader in this area, and if they get first mover advantage, Windows Phone may yet be a competitor, depending on how many other people are interested in Continuum. I think a lot of people might be interested. A lot of people, like me, are sick of babysitting and updating and maintaining and tweaking multiple computing devices.