Though Continuum holds the promise of turning your Windows phone into a near-full PC experience when docked and connected to a display, the feature still falls short in some areas. One of those pitfalls is a little app called Netflix, which hasn't worked with Continuum either through the app or via the Edge browser. Now, at least one of those appears to have changed.
As first spotted by WindowsUnited.de, Netflix will now work over Continuum through the Microsoft Edge browser. If you own a Continuum compatible phone, like the Lumia 950, Alcatel Idol 4S, or HP Elite x3, then you should be able to stream to your heart's content over Continuum — so long as it's through Edge. Unfortunately, the Netflix app still doesn't work over Continuum.
As WindowsUnited points out, streaming also works (and has for some time) for Amazon Prime Video content via Edge. You'll just have to make sure that you have Edge set to desktop mode to see the streamable content.