For my own reference, and anyone else trying to accomplish this: The Dell U4919DW monitor has built-in KVM switching.  Meaning, you can hook up one keyboard and mouse to the monitor and control multiple computers.
Where this becomes interesting is in Picture-by-Picture mode (PBP).  You can have to computers displaying on the same monitor simultaneously.  However, you can only control one at a time using the attached keyboard and mouse.  So how do you easily switch between them?
Until now I had been using the on-screen display (OSD) using buttons on the monitor.  This is fine when really only using one computer at a time, but for rapid / frequent shifting it becomes tedious.  Instead I needed a way to do this from the keyboard.
On a windows machine, you can install Dell Display Manager (or Dell Display and Peripherals Manager) and program hotkeys to switch KVM inputs.
But my non-work computer is linux.  For that you need to install ddcutil ("sudo apt install ddcutil" on ubuntu) and execute: ddcutil setvcp E7 0xFF00
You can alias that command, throw it in a script, or set some hotkey / hot corner operation to facilitate that.
Now I can flip between computers, having both displayed simultaneously, without having to navigate through the clunky OSD
Source: 
https://www.dell.com/community/en/conversations/linux-general/dell-display-manager-keyboard-shortcuts-for-kvm-switch-on-linux-ubuntu-for-u3821dw-monitor/6698ead384a7860c8e48bca0