What Apple ID should I use for my work computer?
There are arguments and reasons to go either way.

First the facts we know (excluding jailbreak and cracking app store DRM):


You can never change the Apple ID for each purchase once made. However, with MDM solutions evolving and VPP, there is the ability to purchase apps in bulk and then revoke the "purchase" by a subsequent AppleID and return a purchase to the pool for re-issue to another AppleID on later iOS versions. Perhaps that will come to the Mac at some future point.
Apple has a Volume Purchase Program to make the App Store work better for business use.
The consumer level Gift Program also exists to let one party pay and another redeem.
You can change the payment option on your account with great ease.
There is a 5 account limit on most devices.
Updating apps is a chore each time you have to:


get updates for account A
go to store preferences
enter account B and the password
go back to the app store for updates 
update again - entering password B a second time
go to store preferences
enter account A and the password

iTunes only backs up apps for which that computer is authorized. (this can be a benefit or a hassle)


Now for some assumptions and commentary. Starting from the simplest scenario, you will either have to use a work credit card or work gift card on your personal Apple ID (juggling the payment options and providing these receipts to account for these purchases just like you would a cup of coffee, a hotel room and other petty expenses that the business just writes off.)

Having a "work" id means in theory - the work could ask you for the password (or force a reset of the password since they would likely maintain control of the email or request/force you to put their email as the contact when you leave the company). They could then issue that Apple ID to another person that would  then have use of the apps you once purchased. In practice - how would you feel getting a "recycled" Apple ID as the new employee or the boss welcoming a new team member. Even $1000 a year in app purchases pales in comparison to the training costs to a business for losing an employee. Company "ownership" of these app purchases is not a big deal in practice.

If you ask your employees to switch in the business account - what happens when someone fails to do that - you'll end up with an expense report anyhow or you'll have a grumpy employee. Banning personal Apple ID is a potential solution as well. That also is demeaning to someone that's going to be taking their work on the go with them - especially if work didn't buy the hardware.

There are times when I really want my work persona to be totally different than my personal. Having that work Apple ID separate from my home ID doesn't seem to work at all to have that separation. I would not be able to remember to switch my account - I would go to purchase - see the password and enter it. About half way through the download I would realize my mistake. I can see this working for some, but think through this if you embark on a two ID strategy.

I would draft an addendum to your normal expense reporting policy to call for quarterly reports (if you don't report these expenses as they happen), urge you to use existing work credit cards to buy gift cards or have one work account gift apps to users that don't need many or are getting started. Once your company hits thousands of dollars of apps - you can look into the Volume Purchase Program which still lets the business send redemption codes to individual Apple ID. Make it a point to educate everyone - an executive summary (this is what we're doing/ how it will be controlled) as well as a detailed guide - who does what and when. 

Once you think it through - the idea of work asking for a work Apple ID might make as little sense to you as it does to me. I would however, be sensitive to the finance department - the have long held practices on how to purchase software. Sitting down with them and showing them my purchase history and educating them how this all works. Handing them my phone and letting them buy an app - seeing it download and start using it is an eye opener.