I have a CC account from my job that I use for InDesign on both my home and office computers. I want to purchase a separate photography account for personal use, primarily for Lightroom/Photoshop, but I need to continue to use both the personal and work accounts on my home machine. Is that possible? Will I need to log out of the work account in order to access the personal?
Moving to Adobe Creative Cloud
You are right if you subscribe to different apps using different Adobe ID you will need to sign out of 1 account from your Adobe CC desktop app and sign in with the other one to use your other apps.
Please refer Activation / Deactivation / Sign in / Sign out FAQ: How do I activate my Creative Cloud Apps? and Activation / Deactivation / Sign in / Sign out FAQ: How do I deactivate Creative Cloud from a computer?
Hope that helps!