Your Pure's serial number should be visible in two places in Mudita Center, including the screen that appears after you connect the phone.

Serial Number Easily Accessible

Open Mudita Center and connect your Pure to your computer with the included USB-C cable. After the application is done syncing with your phone, it should display the "Overview" section. Your phone's serial number will be visible right under the photo showing the device, above the "Disconnect" button.

The serial number is also displayed in the "About your Pure" section, which you can enter by clicking on "About your Pure" below the "Disconnect" button in the Overview.