Yes, it already works with watchOS 3 and iPhones with built-in barometers (iPhone 6, 6 Plus, 6s, 6s Plus, 7, 7 Plus, 8, 8 Plus, X) running iOS 10.x or newer.
Start an outdoor walk workout and walk uphill. Once you stop the workout you will see the elevation gain on the Apple Watch and in your iPhone's Activity.app.
iSmoothRun records elevation data using the iPhone's barometric altimeter. This app can sync with various storage and tracking services, such as Dropbox, RunKeeper, Strava, etc.
The barometric data is preserved when syncing or exporting GPX/TCX files from iSmoothRun (although I currently have to tweak the header of the GPX file and manually upload to Strava ...
Barometer++ (full disclosure: I worked on the app at Friends of The Web) shows the raw pressure values from the iPhone 6 or 6 Plus's barometer sensor. It also shows sea-level adjusted pressure to compare to weather stations around the world (using your GPS altitude), and shows elevation changes based on the barometer.