I have not done a thorough study of this verse, but I would make a couple of general comments.
First, Hebrews 1:3 clearly states that Jesus Christ is not just like God, but is God. Jesus is the full and perfect image or representation of God. Furthermore, it states that Jesus Christ (verse 2) made the world, and (verse 3) sustains it (upholds all things by the word of His power). In other words, He is not only the representation of God; He does (and has done) the work of God.
Second, while Jesus Christ is God, He is the One who made purification for sins (verse 3), and sat down at the “right hand of the Majesty on high.” Thus, Jesus is distinct from the Father, for the Father sent the Son, and He sits next to the Father. The Father and the Son (Jesus Christ) are both God, but they are distinct.
That is a truth that boggles the mind, but it is exactly what the doctrine of the Trinity maintains.