When performing a "Check for Updates" a dialogue box persists and must be manually removed from the Dock. Refer to screenshots for details.
Version details - screeenshot 4
@ausernie - sounds like you're clicking the 1Password updater in macOS 10.14 (“Mojave”)'s new "Recent Items" feature in your Mac's dock, is that correct? If so, the best strategy is...don't click that.
The way to check for updates to the software in 1Password hasn't changed -- either set it to check automatically for you or check manually by clicking 1Password's Preferences > Updates > Check Now. What you're seeing in the Dock is the actual updater application which is launched by 1Password to download an update once one is found. In previous (pre-Mojave) setups, you would never see the application icon for this updater utility, because it was only present to download the update: if 1Password found a new version, it would call this updater which would download the new version, quit 1Password, install the new version and then quit.
That's still what happens, but because this little updater utility is a separate tiny application from the main 1Password, Mojave's "Recent Items" feature picks it up if it was recently used. If you open other applications, you will notice that you no longer see the 1Password Updater in that position in the Dock, because Mojave only displays the three most-recently opened apps...so it gets "bumped" if you keep opening other apps. Another option would be to visit System Preferences > Dock and UN-check the button marked "Show recent applications in the Dock." That way you won't see this - or any other - "recently used" applications, and regain a little of your Dock space. And, as I mentioned, the best method of ensuring this doesn't happen is to simply not click the updater when you see it there; continue checking for updates in the way you have always done instead in 1Password. Hope this helps!