Yes, there are several apps and tools available that allow you to transfer messages between your desktop and mobile devices. Here are a few options:
Pushbullet: Pushbullet is a popular app that lets you easily send text messages, links, files, and more between your desktop and mobile devices. It also has a notification mirroring feature that lets you see your phone's notifications on your desktop.
AirDroid: AirDroid is another app that allows you to transfer messages, files, and other content between your desktop and mobile devices. It also lets you remotely access and control your phone from your computer.
Join: Join is a Chrome extension that lets you send and receive SMS messages, links, files, and more between your desktop and mobile devices. It also has a notification mirroring feature similar to Pushbullet.
MightyText: MightyText is an app that lets you send and receive SMS and MMS messages, as well as view your phone's call logs and contacts, from your desktop. It also has a notification mirroring feature and supports group messaging.
All of these options are available for both Android and iOS devices.