It is common to have two equally separated windows by either dragging each window to one side of the screen or Ctrl+Super+←/→, but I want to divide my screen into four quarters.
I have encounterd many situations in which I had to switch windows between three or four. It would be sometimes easier to debug any code if I could have a screen with four equally divided windows so that I could see the entire system with one look.
It is possible, as you can see from this example with four gnome-terminal windows:
I did not manually resize and position the windows, which I could have done, but just dragged them into the corners of the screen to format them automatically. It's the same procedure as you described with placing a window into the left or right half by dragging it to the respective border of the screen. There are also custom keyboard short-cuts to achieve this.
You can set up whichever method you prefer or both.
However, the corners are by default configured the same way as the side borders and the short-cuts are probably disabled, so we have to change these settings.
Therefore we need the CompizConfig Settings Manager 
.
You can also install it from the terminal with the command below:
sudo apt-get install compizconfig-settings-manager
After that, launch ccsm.
Navigate to the category Window Management and select the Grid plug-in. Go to the Corners / Edges tab and you should see the following settings:
Set all corner options according to my example above, so that when you drag a window e.g. into the upper left corner, it will get resized and placed into the top left quarter of the screen.
You can also set up keyboard short-cuts to place windows into the grid. Switch to the Bindings tab and select the short-cut key combinations you prefer. I opted for Ctrl+Alt+NumPad 1-9 to put the windows into the respective screen halves or corners. See my example below:
All your changes take effect immediately, so just close the ccsm window.
Now you can start dragging windows into the corners:
You can do this as well with the Unity Tweak Tool.
Install it with the terminal command below or by clicking here: unity-tweak-tool
sudo apt-get install unity-tweak-tool -y
Once you installed this tool, open it and go to Window Snapping and then you can configure the window snapping on Ubuntu.
All the described solutions do not work with Ubuntu 18 and up except ShellTile.
You can use the gTile gnome extension. This is my preferred solution which has more features. These are the default shortcuts:
Read the github page for more shortcuts and settings.
There is a little application called x-tile that can organize windows in several different ways, including the way you want (tile all quad). Make sure to manually install python-appindicator (missing depency -- filed a bug for it).
The generic keyboard settings dialogue allows shortcuts to be set for view split on left or right.
Additional shortcuts can be added with the + button, although I do not know what the commands would be.
A solution that worked out-of-the-box for me is WinTile. It's a GNOME extension that imitates shortcuts from Windows (basically Super + <Arrow> to move / resize a window). Window snapping is supported as well, for a variable number of columbns and rows.
