One of the few agreements between all 4 of the Gospels: Matthew Mark, Luke, and John, is that the public ministry of Jesus begins with the proclamations / testimony of John the Baptist about Jesus. While the exact wording of John's testimony varies between them, all point to Jesus as greater than John.
John, however, does not describe the baptism of Jesus. Instead, John the Baptist testifies:
I saw the Spirit descending on him like a dove, and it remained on him. I myself did not know him, but the one who sent me to baptize with water said to me, "He on whom you see the Spirit descend and remain is the one who baptizes with the Holy Spirit." And I myself have seen and have testified that this is the Son of God.
John 1:32-34, New Revised Standard Version
In this description, the context in which John the Baptist sees the dove descending is not given. Was Jesus baptized? Did Jesus see the heavens open? Did Jesus hear a voice declaring, "This is my Son?" The Gospel of John is silent on all these points.
In Matthew, Mark and Luke, the sequence of events is: Jesus is baptized; the heavens open; the Spirit descends on Jesus like a dove; a voice says, "This is my son, the Beloved, with you (or, with whom) I am well pleased."
In Matthew and Mark, the heavens are opened "to him," and "he sees" the Spirit of God descending; suggesting that perhaps these experiences were private to Jesus alone.
It is unclear whether the voice from Heaven would have been heard by all those present, but such an affirmation of status and honour would make no sense unless it were heard publicly. Indeed, in some ways, it is possible to read the rest of the account of Jesus' life as being public confirmation of what is announced at Jesus' baptism.
(Aside: an interesting Christian heresy is to propose that Jesus is adopted as God's son at his baptism. Why is this interesting? Because adoption is a key metaphor for our relationship with God.)
Only in Matthew does John protest baptizing Jesus (Verse 3:14). After all, since John is baptizing with water for repentance (Verse 13:11), what need does Jesus have for being baptized? This protest of John is similar to Peter's protest when Jesus first reveals that he will suffer and die (Verse 16:22).
And Jesus' response here:
Let it be so for now; for it is proper for us in this way to fulfill all righteousness.
foreshadows his prayer in the Garden of Gethsemane before his arrest and execution:
My Father, if this cannot pass unless I drink it, your will be done.
That is, just as the execution of Jesus on a cross confounds all Messianic expectations at the end of his life; so too, his public ministry begins with a confounding of social norms. The fulfilling / completing of righteousness requires letting go of our personal sense of what is right and proper, of our own will, in order to fulfill God's will.
As at the end, so too at the beginning, a deep and profound abiding in / aligning with the will and Spirit of God.
* Link to Amazon.com Bibliography for Bruce J. Malina and Richard Rohrbaugh, Social Science Commentary on ... The Synoptic Gospels; The Gospel of John; The Book of Acts; The Letters of Paul; The Book of Revelation; and others.
+ Link to Amazon.com Bibliography for Amy-Jill Levine and Marc Zvi Brettler, Jewish Annotated New Testament, The Bible With and Without Jesus, Short Stories by Jesus, Entering the Passion of Jesus, and others.