Coat of arms for the Anglican Diocese of Trinidad & Tobago
I remember being vilified by one particular self-identified Catholic who said the Church teaches that God is one Person in three modes and I was wrong, thus contradicting priests. I have no idea what he is talking about. For all I know he could be lying about being Catholic, or misunderstands what his priest says or even his priest is preaching heresy. What he was telling me is called modalism or Sabellianism as it is named after a 3rd century Catholic priest named Sabellius who disagreed that God was one in three Persons: Father, Son and Holy Spirit.
Saint John Paul II wrote that the Trinity is “one God in three persons” . As the Eleventh Council of Toledo said, “He is not the Father who is the Son, nor is the Son He who is the Father, nor is the Holy Spirit He who is the Father or the Son” . Saint Irenaeus of Lyons wrote, “Now man is a mixed organization of soul and flesh, who was formed after the likeness of God, and molded by His hands, that is, by the Son and Holy Spirit, to whom also He said, ‘Let Us make man.'” . Saint Theophilus of Antioch wrote this.
“Moreover, God is found, as if needing help, to say, ‘Let Us make man in Our image, after Our likeness.’ But to no one else than to His own Word and wisdom did He say, ‘Let Us make.'” 
In both cases we see a clear doctrine of the plurality of God as the Father says to His Word and His Spirit, “Let us make man in our own image” . In fact, the Israelites called God Elohim which is plural for El. He is called Elohim in the Masoretic text of Genesis 1 and 2. This is significant in that if God were one Person in three modes, then there would be no reason to call Him ‘divine beings’ or Elohim. Rabbinical Judaism accepts the henotheistic belief that God is speaking to His angels in the creation story and that Elohim just expresses this idea that God is great, but henotheism was rejected by the Old Covenant as idolatry, as the Lord God Almighty said, “You shall have no gods besides me” . Maimonides popularized the concept that “Elohim is a homonym, and denotes God, angels, judges, and the rulers of countries” , but that was far later in the Middle Ages.
Saint Justin Martyr rejected henotheism saying, “God speaks in the creation of man with the very same design, in the following words: ‘Let Us make man after our image and likeness…And God created man: after the image of God did He create him; male and female created He them.’…For I would not say that the dogma of that heresy which is said to be among you is true, or that the teachers of it can prove that [God] spoke to angels, or that the human frame was the workmanship of angels” , but he said, “He whom Solomon calls Wisdom, was begotten as a Beginning before all His creatures”  and this Wisdom cooperated with God the Father in creation. So, now it has been established that the Elohim mentioned can neither be one Person, nor can He be angels and judges as they, especially judges, could not have been in the beginning with God and angels are created beings of God (rf. Psalm 148:2.5). How could the Son pray to Himself in the Garden, or send Himself to His apostles when they were already with Him, or be sent from Himself to Earth? Obviously plurality of Persons is logical.
 Catechism of the Catholic Church, no. 253
 Council of Toledo XI (675 A.D.), no. 314
 Saint Irenaeus of Lyons, Against Heresies 4:4
 Saint Theophilus of Antioch, To Autolycus 2:18
 Genesis 1:26
 Exodus 20:3
 Maimonides, A Guide for the Perplexed, p. 14. <http://www.sacred-texts.com/jud/gfp/gfp012.htm>.
 Saint Justin Martyr, Dialogue with Trypho, ch. 62