The problem with reconciling such a belief with the orthodoxy of the Church was that the Catholic interpretation of God as the Trinity, while it places the Father, Son, and Holy Spirit as co-eternal, equal beings, does not go so far as to say that all three are the same person, which is the principal tenet of the flavor of Monarchism known as Modalism, which states that Jesus and the Holy Spirit are exactly one and the same as the Father (in different "modes").
The other flavor, Adoptionism, has a very different slant. While it still holds that God is one being, above all else and wholly indivisible and of one nature, it reconciles the "problem" of the Trinity (or at least Jesus) by holding that the Son was not co-eternal with the Father, and that Jesus essentially was granted deity-hood (adopted) for the plans of God and his own perfect life and works. Different flavors of Adoptionism hold that Jesus was "adopted" either at the time of his baptism, or ascension. An early exponent of this belief was Theodotus of Byzantium.
Thus, Monarchianism in-and-of itself does not represent a complete theory of the relation of the Father, Son, and Holy Spirit so much as a simple tenet that requires further extension.