First, I would strongly suggest that you have a work computer and a personal computer, and then keep those two separate for legal reasons. Although this is not the place for legal advice, and there are many other factors to consider, you should know that in general:

1. Your employer owns your work computer, and can legally confiscate it at any time and for any reason. Thus, you should consider any personal information you have on your work computer to be accessible by your employer. This includes personal information like tax forms and private correspondence. It also includes information you might not want your employer to have, like criticisms of the administration or job offers from other institutions.

2. Academics tend to have many varied endeavors inside and outside of their academic profession. Your employer probably has a very strong claim to the intellectual property rights of anything you create on their computer, even if the IP does not relate to your university job and even if you're only using generic software such as Microsoft Word.

3. Your position at a university may expose you to FERPA or HIPAA protected information, and your university may have specific expectations about how you access that data. My university insists that all laptops use whole-disk encryption because the loss or theft of unencrypted student records is a major FERPA event that must be disclosed to the government and/or public.

Second, there are some practical and legal problems with retaining your own computer from a software licensing point of view.

1. There's a high probability that a lot of the software on your current computer should no longer be used according to common academic licensing agreements. This is definitely true for certain specific software such as MATLAB, which are generally licensed to the university for use by university students and employees (this is called a "site license"). Since you are no longer a student of that university the license demands that you stop using that software.

This all depends on the specific licenses that your university has negotiated, but the situation above is very common. It almost certainly applies to any proprietary technical software you've got, and there's a very good chance that it also applies to any proprietary productivity software you've got (e.g. the Microsoft Office suite, VPN software, etc.). It may in some cases apply to the operating system itself, though this is less common today than it used to be.

2. The reverse of the above situation is also a problem. Many university licenses stipulate that software may only be installed on university owned computers or on student computers. As a postdoc you're no longer a student, so their IT department might balk at installing any work-related software on your personal computer.

First, I would strongly suggest that you have a work computer and a personal computer, and then keep those two separate for legal reasons. Although this is not the place for legal advice, and there are many other factors to consider, you should know that in general:

1. Your employer owns your work computer, and can legally confiscate it at any time and for any reason. Thus, you should consider any personal information you have on your work computer to be accessible by your employer. This includes personal information like tax forms and private correspondence. It also includes information you might not want your employer to have, like criticisms of the administration or job offers from other institutions.

2. Academics tend to have many varied endeavors inside and outside of their academic profession. Your employer probably has a very strong claim to the intellectual property rights of anything you create on their computer, even if the IP does not relate to your university job and even if you're only using generic software such as Microsoft Word.

3. Your position at a university may expose you to FERPA or HIPAA protected information, and your university may have specific expectations about how you access that data. My university insists that all laptops use whole-disk encryption because the loss or theft of unencrypted student records is a major FERPA event that must be disclosed to the government and/or public.

Second, there are some practical and legal problems with retaining your own computer from a software licensing point of view.

1. There's a high probability that a lot of the software on your current computer should no longer be used according to common academic licensing agreements. This is definitely true for certain specific software such as MATLAB, which are generally licensed to the university for use by university students and employees (this is called a "site license"). Since you are no longer a student of that university the license demands that you stop using that software.

   This all depends on the specific licenses that your university has negotiated, but the situation above is very common. It almost certainly applies to any proprietary technical software you've got, and there's a very good chance that it also applies to any proprietary productivity software you've got (e.g. the Microsoft Office suite, VPN software, etc.). It may in some cases apply to the operating system itself, though this is less common today than it used to be.

2. The reverse of the above situation is also a problem. Many university licenses stipulate that software may only be installed on university owned computers or on student computers. As a postdoc you're no longer a student, so their IT department might balk at installing any work-related software on your personal computer.