Android Messages on desktop is seemingly just around the corner. It appears that Google has completed the development for the feature within the Android app and the screenshots below give you a glimpse at what the option will look like when going into the app to set it up. Still absent is the actual desktop side of the feature in regards to it being live, but the website (listed as messages.android.com) where you'll go to set everything up in your browser does appear to be up, though it does only lead to a 404 error message at the moment so you can't actually configure anything.
Though you can't actually set the feature up yet to allow messaging from your PC, once Google turns things on you'll be able to start messaging from your desktop browser by scanning a QR code in the PC screen just like you do with Allo. The option to set everything up within the app side of things will be found in the overflow menu, and after Google pushes out the feature you should see a prompt about the capability come up from the bottom of the screen the next time you open the app.
Though the Android portion of the development may be finished and Google may just have to turn things on server-side, there's still no indication of how long it will be before the feature does go live. That said this is about as much proof as anyone would need that Android Messages on desktop is eventually coming, it's just a matter of when. Google account linking also seems to be a feature of the Android Messages for desktop, which Google lists as enabling a "more personalized experience" though doesn't elaborate on what that means. It's possible that it will be an option for users to add if they want Messages on the web to allow for more accurate predictions and auto-fill, though it could also be for something else entirely.