Thank you ! I must admit I count the one in the east as part of the same empire - they still described themselves as the Roman Empire, and despite speaking Greek thought of themselves as a continuation of the same entity.
It's an interesting thought experiment to wonder if Legio V Macedonica had a vexillation or cadre unit back in Dacia Ripensis which survived after the part in Egypt disappeared during the Muslim conquest. Theoretically they could have survived the various reforms of the army, becoming some kind of Thematic or Tagmatic unit, then skutatoi. Although admittedly unlikely, it's possible that one of the units defending Constantinople in 1453 could have traced it's origin right back to 43BC 😀
Restoring the Republic seems to have been a consistent theme among various emperors, but I have a suspicion that they were just paying lip service to it as a way of harking back to an idealised past (the Romans did that a lot !) Nominally the Republic and it's structures still existed, but most people knew where the real power lay, and recognised how bad the Republic had become towards the end (once Marius and Sulla began the pattern of dictators ruling).