Right click on a Shulker Monster while holding a Chest, and it transforms into a Shulker Chest Monster.
It has a 27 slot inventory.
It is still hostile, sort of.
When right clicked with an empty hand, a GUI for it's inventory opens, and it temporarily stops attacking you, but still attacks other players. Exiting the GUI allows it to start attacking again.
The monster's inventory can be loaded and unloaded using hoppers, very much like a minecart with a hopper.
Where a normal Shulker Monster teleports completely randomly within a 17x17x17 cube centered on itself, the Shulker Chest Monster will first attempt to teleport 8 blocks away in each possible straight line direction (in random order), and if that fails, it will attempt to teleport normally (randomly).
Shulker Chest Monsters are intended to be used as a form of item transport, obviously. Because they prefer to move in straight lines, you can control their movement by ensuring that one of the six blocks it will first try, is air and has a solid block next to it. A piston should probably suffice to provoke them into teleporting for you, either triggered by a redstone clock, or a tripwire circuit.
In terms of speed of getting items from one area to another, a Shulker Chest Monster based item transportation system should have a lower latency (meters/second) than anything in vanilla.
Obviously nothing can match the throughput (items/second) of items in a water stream, as you are limited to the speed of loading/unloading.
Like any other non-boss mob, you can transport them with minecarts or boats without them attacking you. This is of course slower, but safer, too.
Double Shulker Chest Monster if two are sitting side-by-side?