First, it is worth knowing what Jung's psychology was based upon, and then it might become clear whether there was anything 'Christian' about it. I quote from a scholarly article on this:
"Carl Jung was fixated on the symbolic parallels between external
'things' and internal mind/consciousness. He understood himself not as
a psychologist, but as a psychological sorcerer. His occult beliefs
concerned a supposed unity of good and evil in the human psyche and in
God. His occult practices were what New Age people today would
recognise as "channeling". He was a pantheist. He taught that we have
to accept evil as the partner of good. That is his idea of
"integrating" evil with good.
He admitted that his "dreams and visions" formed "the prima material
of [his] scientific work." He was channeling, making contact with
spirit guides. His belief in synchronicity meant that he corresponded
interior or mental things and events with exterior or physical things
or events. This led him to believe that the world has no reality apart
from the mind that experiences it. This enabled him to dismiss the
question whether his personal myth is true. He believed it made no
difference whether his gurus or guides were real spirits or just
fantasies. But when he described the way they affected him, he used
adjectives like "demonic" and "diabolical".
The frankest of his confessions are in his "Psychology and Religion :
West and East". Also in his "Memories, Dreams, Reflections". He
spoke of entities called Elijah, Salome and Philemon (who was a pagan
and brought with him an Egypto-Hellenistic atmosphere with a Gnostic
coloration.)" Extracted from an article by Prof. J Budziszewski in
"Christian Research Journal" Spring 2000, pages 30-33 http://www.equip.org
Many people who are unaware of the extent to which Jungian beliefs have informed today's atheism and other worldviews might be surprised to discover the extent to which popular culture has imbibed many Jungian ideas; Star Wars ('the Force'), Star Trek (one early episode depicting the good force as listless, frail and irresolute), The Dark Crystal (a children's film).
Now we may be better enabled to respond to comparing Jung's 'ego death' with the biblical concept of "death to sin". My point will come from Paul's words in 1 Timothy 1:15 - "Christ Jesus came into the world to save sinners: of whom I am chief".
Paul's use of present tense (as opposed to saying 'of whom I was chief') accords with Christian belief that becoming a Christian does not stop you from continuing to sin, and that although God does not hold your past sins against you, that does not erase the fact that you committed them. Anyone who knows what Paul did when known as Saul of Tarsus will understand his meaning and his immeasurable regret at those sins. Context shows he was using himself as an example of the extent of Christ's amazing saving power. But Jung's esoteric understanding is utterly at odds with the reality of actual sin committed by people, and how God deals with what they truly did, via a truly historic event witnessed by many on a dark hillside called Golgotha, nearly 2,000 years ago, outside of Jerusalem.
The answer submitted is that there is nothing similar whatsoever between Jung's 'ego death', and Christian (i.e. biblical) 'dying to sin.'