There is also the issue that inasmuch that Peter was the chief of the Apostles, his successors were not only in Rome, but primarily in Antioch and every other place. Alexandria can even claim Peter as its founder, as he sent St. Mark there, and consecrated him as Bishop over the Church there. Constantinople itself received its territory from Antioch when it became the center of the Empire. The whole nobility and government of Rome moved to Constantinople. Lastly, the little fishing village of Byzantium, which had a Bishop there from the time of St. Andrew the Apostle, and St. Stachys of the 79 and first bishop in Byzantium. The city became known as New Rome officially, as much as Old Rome was the capitol of the pagan empire, now New Rome was the capitol of 'Christendom'. Rome retained its Bishop as the Patriarch over the Church in the West, much as Antioch continued over that in the East, Alexandria in the South, and now Constantinople in the North.
If they followed the argument through to its logical conclusion, then the Pope of Alexandria, or the Patriarch of Antioch (called the 13th Apostle) would have the same powers as the Pope of Old Rome. Either way, the Petrine office is not Rome's alone, and much of her honour due to the Empire passed to Constantinople when the Empire became Christian.
Ari, sinner