Want to keep certain conversations private on the Messages app on my Mac

I have an iPhone and I use a Mac computer at work. I have my iPhone synced to the messages app on my Mac so I can send and receive texts on my computer. However, since it is a work computer I would like to be able to keep certain conversations private to just my phone so they would not show up in the messages app where someone else could read them. Is there a way to do this in the messages app or is there another app I could use? I talked to Apple support and they suggested the Do Not Disturb feature. However, this only hides notifications from that conversation but the conversation still shows up in the app. Obviously a workaround would be to just delete those conversations whenever I am going to be away from my computer or just not have them sync but I want to see if there is a better option out there. Any insight would be greatly appreciated. Thanks!