It's my understanding that "My messenger" refers to John the Baptist, and the "Me" refers to Jesus, the Messiah, the messenger of the covenant. My notes have a cross-reference to Psalm 85:8,
"I will hear what God the Lord will speak, for He will speak peace to His people and to His saints . . ."
I can tell the latter part of this verse is covenant language. Can anyone tell more about this?