This is exactly why so many of my friends and family use Hangouts née Gmail Chat.
We can use it in the browser at work just by keeping Gmail open in a tab. No IM applications to install or concerns about whether we're running Windows, OSX, or whatever.
Then the conversations follow when you log off whatever desktop or laptop you're on via the mobile app. About half of us use Android so it either comes standard or you can easily install. The other half uses iOS and unlike iMessage/iChat/Facetime, you can easily use it on iOS as well as Android.
I installed Allo today but so far I haven't had a chance to try it out. Frankly it gives me the same mild annoyance I felt when Facebook broke messaging out into a separate app (and when I just said screw it and uninstalled both).
As much as I use and generally like Google stuff, they drive me nuts with their approach a lot of the time. When GChat/Talk became Hangouts and started integrating Voice and SMS/MMS, it seemed like a move in the right direction, both for them as a company (wanting to be your one stop shop for messaging) and for me as a user (less apps, more seamless integration). Then SMS/MMS was split back off into Messaging, Voice...I haven't really heard much about Voice in a while which is weird because it was one of those things that seemed so promising at one point. And now Duo alongside the pretty great video chat already in Hangouts and Allo alongside Messaging alongside Hangouts.
I get the idea of maybe starting a secondary platform/service/app to see what sticks and what doesn't. At worst they can integrate the good bits into their main product (see: Inbox). But their messaging strategy seems more like publishing Inbox with the intention of just ditching plain old Gmail.