Picture of a spaceship occasionally appears on my monitor when switching monitors

If the resolution of your monitor isn't quite set up correctly, your video card shows the correct pixels for your desktop, but then the extra monitor space gets filled with essentially random pixels from elsewhere in your video card's memory. From your comments it sounds like that is usually whatever is on the bottom of the previous display screen. If you've never looked at anything with that particular spaceship, it could be an image that is part of a video card self-test, or it could be a hidden background image from some other webpage.

Fixing the external monitor's resolution is the correct fix for this problem. If you've got other issues after doing that, please submit them as separate questions.