Add a new notification in the Pushcut app and set its name.
Title and message will be set dynamically, so they will not matter.
Tap Default Action and select URL.
In the URL field, tap Input to use the Raw Input as the URL to open. We will later pass in the tweet's link via the notification input.
Tap Done and Save to activate this notification.
Go to Zapier and add a new zap.
Select Twitter as the trigger and choose the triggering event.
This example uses the Search Mention event with pushcut as the term.
Add an action to your zap, select Pushcut and choose Send Notification.
The first time you will be promptet to link your account. Follow the instructions and scan the QR code with the Pushcut app (Account > Scan Code).
Select the Pushcut notification you created before from the drop-down and pass in the tweet's text and username into the notification.
Make sure to pass the URL as the Input Parameter, as it will be used in the default action.
Click Test & Continue and verify that your notification behaves correctly.
When all is set up to your liking, do not forget to toggle the On switch to activate your zap.
That's it! Your twitter notifications will arrive exactly as you define them them.
You can of course use Zapier's power to filter them even more or include different information.
If you prefer a third-party twitter app (like tweetbot, twitterrific, ...), you can use custom URL schemes to open your tweet there.