There was an old feature request similar to this but being over 6 years old and still not being implemented I thought it was worth bringing up again. I've been evaluating DbVizualizer to replace my old Navicat copy which is just way too expensive even just to upgrade and so far have been fairly impressed with the quality of the application for the price, especially not forcing me to buy separate licenses for Mac and Windows (like many devs I work on a Mac but often have to use a Windows VM for some clients due to VPN issues).
But the one feature I most miss from the other tool that is so helpful I am considering still spending the money to get it is a search box that as you type into it, immediately filters the object list to match. When you are working with clients that have hundreds, if not thousands of database tables, views, SPs, etc. you really need a way to quickly locate the object you want to work with, but many times you may not know the exact name, especially with new clients that don't document their databases (far too common in my line of work!)
Using the Type-to-Find is a reasonable way to find it if you know the name off the top of your head, but it doesn't actually filter the list which is what makes the list of objects more manageable, and I don't seem to be able to go to the next result, or easily go back to my results and keep looking if I open a table or SP and find it's not the one I want. While there is a way to filter the database objects, it's time consuming and tedious and best suited for setting up a commonly-used set of filter rules, vs. something you need to do dozens of times a day with a different search value. Having a quick filter box on the menu of the object window that would filter the database objects as you type based on a partial match of the object names is just such a massive timesaver when working with large databases.