Hi, Thomas.
Thank you for visiting Apple Support Communities.
It sounds like you are experiencing an issue with Messages and Continuity. I understand you have made sure that your Apple ID information matches on both devices. Could you make sure that Text Message Forwarding is enabled and your iMac is selected?
Your iPhone, iPad, and iPod touch need to use iOS 8.1, and your Mac needs to use OS X Yosemite.
- Sign in to iMessage on your iPhone, your other iOS devices, and your Mac using the same Apple ID.
- On your iPhone: Go to Settings > Messages > Send & Receive > You Can Be Reached By, and add a check to both your phone number and email address. Then, go to Settings > Messages > Text Message Forwarding, and enable the device(s) you would like to forward messages to.
- Your Mac, iPad, or iPod touch will display a code. Enter this code on your iPhone to verify the SMS feature.
Use Continuity to connect your iPhone, iPad, iPod touch, and Mac
If Messages still are not mirroring on both devices, take a look at the troubleshooting article below.
Get help using Continuity with iOS 8 and OS X Yosemite
Regards