From Carson’s commentary on John 5:19 in his commentary on that gospel in the Pillar series:
“Not only does the Son always do what pleases the Father (8:29), but he can do only what he sees his Father doing. In this sense the relationship between the Father and the Son is not reciprocal.
It is inconceivable that John could say that the Father does only what he sees the Son doing. That would be preposterous not only in the cultural understanding of father-son relationships, but also in John’s understanding of the relationship between Jesus and his heavenly Father (against Gruenler, who tries to make the Father/Son relationship perfectly reciprocal by saying that each ‘defers’ to the other – but this is a ‘fudge’ category that blurs the obvious distinctions).
The Father initiates, sends, commands, commissions, grants; the Son responds, obeys, performs his Father’s will, receives authority. In this sense, the Son is the Father’s agent, though, as John goes on to insists, much more than an agent.” (250-251).
Does Jesus’ submission to the will of the Father make him lower than the Father? Is there some inequality going on here?
Not at all. In 5:18, Jesus’ opponents recognized that Jesus, “was even calling God his own Father, making himself equal with God.”
Jesus doesn’t dispute their accusation, because it’s true. However, he does nuance it, explaining how his actions neither replace the Father, nor take place outside the Father’s intentions.
(I’ve never heard anyone refer to this passage to argue for complementarianism, but it seems to be a nice expansion of 1 Cor. 11:3.)