What is "Christ Consciousness", and which branches teach it?

The first thing to understand about “Christ Consciousness” is that it is not Christian and does not form any part of Protestant or Baptist theology that I am aware of. Below are partial quotes from the official Christ Consciousness Organisation:

Christ Consciousness is the growing human recognition and blending of the human evolutionary (or ego) mind with the Divine Mind and the Divine Personality that is the source of human happiness and fulfillment. This awareness accrues over time within the consciousness of human thinking when intention, attention, and openness is focused on knowing who and what is that “Christed” state of being—the higher mindedness of enlightenment.

Is Christ Consciousness Christian? This path is open to anyone regardless of their religious tradition if and when he or she is open to become a living vessel of LOVE, TRUTH, BEAUTY and GOODNESS on the planet and actively strives to attain it... This is not a term used exclusively in the Christian religion, nor does it mean that you must adhere to a Christian belief system to attain this state. All of the world’s religious traditions offer a path to achieving this “Christed” status, and people are free to find their way in the context of their religious choice. Sources: https://www.institutechristconsciousness.org/ https://www.ctrforchristcon.org/living-christ-consciousness.html

In other words, it doesn't matter if you are a Christian or not, or if you follow another religion, or if you have no religious beliefs, because "Christ Consciousness" is not about believing in and following the teaching of Jesus or of obeying Him. It's about finding your own path - whatever makes you feel good.

This is what one non-denominational Protestant Christian website has this to say about “Christ Consciousness” and whether or not it is Christian:

So-called “Christ consciousness” has been known by various names in history, such as Jainism, Buddhism, Hinduism, and most Eastern mystical religions. More recently, Deepak Chopra has popularized the “christianized” version of this same pseudo-spirituality. The danger in this latest version of mysticism is the use of Bible verses and Christian terms, which can easily lead astray those who don’t check the Scriptures for themselves.

The basic premise of mysticism is that man can, within himself, transcend physical existence and experience his own goodness as being “one” with the universe, being a god, or existing on whatever higher plane he chooses to believe in. The name of Jesus is merely used as a prop in this latest version of the same idea. Jesus is seen as the “leader” in showing us how to exalt our own inner goodness and, in doing so, make ourselves right with whatever deity we choose to acknowledge. Christ consciousness groups claim that Jesus earned the title “Christ” by perfectly channeling the divine consciousness we can all attain, and they attempt to attribute this philosophy to the Lord Jesus. Those championing this ideology are merely using the name of Jesus as a means of worshiping themselves. They desire to find absolution without repentance, confession of sin, or acknowledging Jesus’ substitutionary death and resurrection (1 Corinthians 15:3–4).

The only “Christ consciousness” the Bible teaches is found in Philippians 2:5–7, “Have the same mindset as Christ Jesus: Who, being in very nature God, did not consider equality with God something to be used to his own advantage; rather, he made himself nothing by taking the very nature of a servant, being made in human likeness.” First Corinthians 2:16 says that, as believers, we “have the mind of Christ.” But this is stated in the context of the spiritual man versus the natural man, who is unable to understand the things of the Spirit (verses 14–15). When we are born again (John 3:3), we receive the Holy Spirit as a gift (Luke 24:49; Acts 2:38). He enables us to comprehend spiritual things (1 Corinthians 2:14), serve Jesus from a heart of love (Mark 7:6; 1 Peter 4:11), and live righteously despite temptations (1 Corinthians 10:13; 1 John 3:8–10). Source: https://www.gotquestions.org/Christ-consciousness.html

Protestant Christianity teaches that we are all sinners and incapable of saving ourselves from the consequences of sin, and that salvation is a free and undeserved gift from God to those who recognise their need, repent and come to saving faith in Christ Jesus. The Bible declares that the ONLY way to God the Father is through Christ Jesus. Any person who rejects the sacrificial death and resurrection of Christ Jesus and thinks they can attain some sort of higher consciousness by virtue of their own efforts is not following Christianity.

...for all have sinned and fall short of the glory of God (Romans 3:23).

For God so loved the world, that he gave his only Son, that whoever believes in him should not perish but have eternal life. For God did not send his Son into the world to condemn the world, but in order that the world might be saved through him. Whoever believes in him is not condemned, but whoever does not believe is condemned already, because he has not believed in the name of the only Son of God (John 3:16-18).

But to all who did receive him, who believed in his name, he gave the right to become children of God (John 1:12).

...if you confess with your mouth that Jesus is Lord and believe in your heart that God raised him from the dead, you will be saved (Romans 10:9).

Religion, whether it is “institutional” or not, saves nobody. Christianity is all about believing in and following Christ Jesus, not some man-made philosophy that embraces some Christian terminology but rejects what God, in Jesus, did to rescue us from our sinful state.