What happens to your hotmail (outlook.com) account when you die depends on a few things, including:
The surest way to know what will happen to your hotmail account is to make a plan and leave any necessary instructions to the people who will be wrapping up your estate.
If you want someone to access to your hotmail account after you die, leave instructions about 1) how to log in to the account and 2) what to do with it. Surprisingly, under Microsoft's "next of kin" process, Microsoft will send survivors a DVD that contains all of the data held in the hotmail account.
This policy contrasts sharply with the policies of other email providers – most will delete an account upon hearing of the account holder's death. A Microsoft Q&A page explains that this next-of-kin policy also applies to executors, personal representatives, and custodians – if they can provide proof of their legal status as the account holder's representative. That page also clarifies that "family member" really only includes spouses, parents, and children.
Also surprising is that Microsoft permits surviving "family members" who have login information to access the deceased person's account:
These policies are good news if you want your survivors to access your account after you die, because they are likely to be successful. And it seems that they would have no obligation to close the account. However, you may not like this policy at all if you don't want your family or personal representatives to access your account.
Because Microsoft provides account access to the families and representatives of deceased account holders, you will have to plan ahead if you want to keep your account private after your death. Here are some options:
Microsoft does not currently allow you to decide what should happen to your account when you die, but it might someday. Tech companies are recognizing that users might not want their accounts deleted, and some are providing tools that allow account holders to decide the fate of their accounts. For example, Google's Inactive Account Manager allows account holders to decide whether to delete the account or pass (some) account information on to survivors after a period of inactivity. Keep an eye out for Microsoft to provide a similar tool.
If you don't make a plan for your hotmail account (and no one hacks into it), your account will be deleted after 12 months of inactivity. If you do not want anyone to have access to your email account, and if you expect that no one will try to access your account, this "do nothing" approach could work well.
However, doing nothing will likely have an undesirable outcome if you want to either 1) provide access to your account, or 2) really make sure that the account remains private. On one hand, unless you leave instructions, the person wrapping up your estate (your executor) will have to deal with Microsoft to get your account data. On the other hand, in some states, your executor may be able to get access even if that's not what you want.
Microsoft is unusual in that it will provide your survivors with your account data – especially because until recently, it has had no legal obligation to do so, even if that data was needed to wrap up a person's estate. However, the law is changing and in some states an executor may – in theory -- be able to get account access even if such access is contrary to the company's policies.
In the few states that have laws granting executors authority to access digital accounts, "account custodians" like Microsoft could be required to provide access to executors. These are new and untested laws, and you can expect most companies to resist providing access to accounts, even in the states where the law requires it to do so.
You can learn more about this, including information about your state's law on Nolo's Digital Assets page.
If you do not make a plan, after your death, your Hotmail account will probably continue to exist for 12 months and then Microsoft will delete it. Or if your executor, spouse, parents, or children request it, they can receive the contents of your account and then the account will be closed.
If you want more control over what happens to your account, either plan for it to be deleted, or tell your loved ones how to log into it and what to do with it when they get in.