Set up background for laptop and plugged monitor
You could try the following as a workaround that may meet your needs. Before proceeding, make sure you are also connected to your external display. Follow these steps:


Open Mission Control
If you haven't previously set up additional desktop spaces before, then click on the + sign at the top right of the screen to create a new desktop space
Now enter the new desktop space by selecting it (via the top of the screen)
Set the background image you would like for the ultra-wide screen on this desktop space
Open Mission Control again
Now enter the first desktop space by selecting it from top centre of screen
Set the background image you would like for your laptop screen on this desktop space


So far, what we've done is created an additional desktop space and allocated a unique image to each as a desktop background.

The trick that may work is to now assign certain apps to the space for the ultra wide screen. This is because if you use an additional screen and have apps running on it you will have an additional space for that screen. 

So, in your case, let's say you usually use the ultra wide screen for iMovie. You can allocate iMovie so that it works on the second desktop space. Here are the basic steps to doing this:


Click on the app in your Dock and keep the mouse button down (if you don't keep it in your Dock, then just launch it first)
From the submenu move the mouse pointer to Options 
You'll see you now have a number of options for where the App should display. In this example you'll want to select Desktop on display x (where x will be a '1' or '2').


The idea behind all this is that if you only use your ultra-wide screen when you're using certain apps, then you can automate some of this workflow. You can also add more apps to work on this screen by default.

Notes: 


For this to work as described, ensure the Mission Control System Preference Pane has Displays have separate Spaces enabled.
This also assumes that Mirror Displays is off in your Displays System Preferences.


Hopefully you'll find something in all of this that works for you (or comes close). For more information on how this all works, see:


OS X El Capitan: Work in multiple spaces
Use Mission Control on your Mac
How to use Mission Control, Expose, Split Screen &amp; Spaces on the Mac