An example that comes to mind is Beaker Browser (https://beakerbrowser.com/), that simplifies exploring and publishing on a peer-to-peer web.
Another example, that I just came across today, is Polarized (https://getpolarized.io/). It seems to embed (or integrate with) a web browser as a feature in a larger personal knowledge base system.
In the very, very distant future, I'm guessing we'll see browsers with VR functionality accessed through a headset or something. Maybe with websites actually resembling 3D worlds akin to the 'cyberspace' fiction of yore.
Something involving AR functionality seems plausible too. Not sure how it'd work, but maybe the sites you're seeing would be like dioramas projected into the real world, akin to Pokemon GO.
But that's probably a while off. Coding VR/AR setups as web apps is still pretty complicated as of right now, and the possibilities for a browser that really depends on them haven't been explored all that much yet.
For the short term... It'll probably be pretty boring by comparison. Typical features from mobile apps and desktop programs will keep making their way into web development (I'm guessing screen snapshots, ambient light mode, more stuff involving the camera and gyro sensors in devices, etc) and eventually the web will start seeing them integrated into the mainstream more and more.
Eg. the "grandad" browser with large text, football results & commentary, civic information and communications tools all lined up and easy to use. The "Kids" flavours with varying levels of helicopter parenting. And a bazillion no-JS software dev versions :P.
Perhaps even with modules. Eg. the "Im going on holidays" module displays a curated list of tools (skyscanner/wikitravel etc) to assist with that. The "Im learning to ride a motorbike" motorbike displays curated youtube content (or pre-trained models that reduce spammy search results) in addition to specialised apps to keep track of progress etc.
Ideas from this effort are going to be trickling into browsers for years to come. For example, if you squint a bit, you might see a conceptual resemblance between the Syphon interpreter in Atlantis and WASM in shipping browsers.
I use Firefox and have no problems with website compatibility. Be the change you want to see.
Huge performance gains in Web Assembly could introduce a new age (Web 4.0 or 5.0?). They could become more like an OS perhaps.