Did Constantine move the capital of the empire to Rome?

No. Actually, Constantine moved the capital from Rome to Byzantium, which he re-named Nova Roma ("New Rome"). After his death, the city was renamed Constantinople in his honor. Today it is known as Istanbul.

Actually that is incorrect. Because the Roman Empire was too large to govern, The Empire was split in two-the western part and the eastern part. The western part's capital was Rome, and the eastern's was Byzantium. Rome remained capital of the western part, although the western part declined while the eastern flourished. Byzantium was later renamed Constantinople in order to honur Constantine.

Rome was not the capital of the west. It was Milan. Diocletian subdivided the empire into a western part and an eastern part. He was in charge of the east and co-emperor Maximian the west. Milan was turned into the imperial capital of the west and Nicomedia (in northwestern Turkey into the imperial capital of the east. Rome became the nominal capital of the empire as a whole. Constantine moved the capital of the east from Nicomedia to Byzantium, which he redeveloped renamed either New Rome or Constantinople (some historians argue that it was called Constantinople from the beginning of its becoming a capital and that New Rome was only an honorific title for the city). The western part begun to decline some 70 years after the creation of Constantinople.