I made the above video as an attempt to imagine what a new app picker would look like in iOS on the iPad. Most would agree the existing solution is less than ideal and needs work.
The first change is with the app switcher that is triggered with a swipe in from the right side of the screen. This currently brings up a list of recently used apps, but the interface is not a very good use of that space, and would be better suited as a place where you can put apps and sort them into a folder-like hierarchy for easy navigation. Tapping on one could bring it full screen, while dragging it to a side or corner of the screen would put it in that location.
In the video example, Microsoft Word is running on half the screen, and the user pulls out the app drawer to put Notes in the bottom right and Safari in the top right.
Finally, the mockup has a dock available at the bottom of the screen. The icons would play by the same rules here, and in this case the user simply tapped the Ulysses icon, and all other apps slid off screen to make room for it to go fullscreen.
There is plenty of UI left out of this mockup, and there are many other parts of the interface that would need to be worked out, but when I think about what a version of iOS that can replace my Mac would look like, it's something like this.