I think the cost of entry will have a huge impact on the usefulness of Continuum for Windows Phone. For the sake of argument, let's say that you already have a Continuum capable phone.
First you'll need a spare monitor, likely one that has a HDMI port since you'll need some way to connect a cord or wireless adapter. In the U.S. that's almost a $100 minimum for a HDMI equipped monitor. Perhaps you could use your current desktop monitor but at that point you're already using a full blown PC thus defeating the point.
Then of course you'll need a bluetooth mouse and keyboard. Go cheap and (sans a sale) you can get both for $30 bucks. Now you're at $130 for three separate pieces of hardware. That's not an insane amount of money but I could see why someone would rather invest that $130 into a new laptop, especially when companies like HP are selling new laptops for $200. And let's not get into the logistics of trying to backup files onto an external drive.
Continuum is great for people that are tired of having a desktop, phone and tablet. It's hardware nausea. Speaking personally, it gets a little tiresome connecting cables so I can transfer files from my phone-to-computer-to-external drive and vice versa. I think it would be neat if Microsoft found a way to make Continuum work by connecting your phone to a laptop. IMO, that opens up a lot of possibilities. Cheap, low-powered or old laptops would get a new lease on life.
You could turn something like the
HP Stream 11 into a true all-in-one device. Imagine being able to open your laptop, work completely from your phone and be able to switch to full Windows just in case. And without having to juggle a single cable or a keyboard, mouse, monitor combo. That would be pretty awesome, IMO, though I doubt Continuum will ever work like that.