Yes of course! This isn't as crazy as it seems.
A Private Chat with yourself is a great place to store information only you can access. It is useful for taking notes and making shopping or todo lists or even for starting that first novel.
1) Click on the new chat button to start a new Private Chat:
2) Hit start (don't add anyone else to it!)
3). That's it. Hey presto, you can chat to yourself!
You may rename it or set an icon just like any other Private Chat. If you decide to open your thoughts up to a wider audience you can even add other people (or bots!) to it in the future.