That’s an excellent question. The connection is incredibly deep, as Carl Jung was profoundly influenced by Eastern philosophy, including the Upanishads. He saw his own psychological discoveries as a Western articulation of ancient truths described in Eastern texts. 
Here is a breakdown of how Jung’s concepts relate to the Upanishads.
1. The Collective Unconscious and Brahman
The most significant parallel is between Jung’s collective unconscious and the Hindu concept of Brahman. 
• Brahman (Upanishads): This is the ultimate, all-pervading, universal consciousness. It is the single, indivisible reality from which all things originate and to which they all return. It’s the “ocean” of which all individual beings are “waves.” 
• Collective Unconscious (Jung): This is a universal, inherited layer of the human psyche, distinct from the personal unconscious (your own repressed memories). Jung described it as a “psychic repository” shared by all humans, containing the memories and symbolic patterns of our ancestors. 
The Connection: Both concepts describe a single, universal, and non-personal foundation of consciousness that connects all individuals. Jung’s collective unconscious is the psychological equivalent of the metaphysical Brahman. 
2. The ‘Self’ Archetype and Ātman
The ultimate goal in both systems is the realization of a true, unified Self. 
• Ātman (Upanishads): This is the individual soul or the “self.” The core teaching of the Upanishads (in Advaita Vedanta) is that the individual self (Ātman) is identical to the universal consciousness (Brahman). The great sayings “Tat Tvam Asi” (That Thou Art) and “Aham Brahmāsmi” (I am Brahman) point to this realization. 
• The Self (Jung): This is the central archetype and the goal of Jung’s process of individuation. It represents the unified, whole psyche, achieved by integrating the conscious mind (the “ego”) with all the parts of the unconscious (including the Shadow, Anima/Animus, etc.). It is the archetype of wholeness and totality. 
The Connection: Jung’s Individuation (the journey of the ego to realize and integrate the whole Self) is a psychological parallel to the spiritual journey of Moksha (liberation) in the Upanishads, where the individual Ātman realizes it is not separate from the universal Brahman. 
3. Archetypes and Hindu Deities
The “contents” of the collective unconscious are the archetypes, which find a clear parallel in the figures of Hindu mythology. 
• Archetypes (Jung): These are the innate, universal patterns or “blueprints” within the collective unconscious. They manifest as recurring symbols, themes, and characters in our dreams, myths, and religions—such as the Hero, the Wise Old Man, the Great Mother, and the Shadow. 
• Hindu Deities (Upanishads & Puranas): The Hindu pantheon is a rich, complex expression of these very patterns. Jung saw these deities not as mere historical or religious figures, but as profound psychological symbols—personifications of the archetypal forces within the human psyche. 
• Shiva, for example, can represent the Shadow archetype (the destroyer) but also the Wise Old Man (the ascetic). 
• Kali is a perfect image of the “Terrible Mother” aspect of the Great Mother archetype, representing the destructive and transformative power of nature and the unconscious. 
4. The Persona and Māyā
Both systems recognize that the “self” we present to the world is not the “true” Self.
• Māyā (Upanishads): This is the veil of illusion—the everyday, phenomenal world that we mistake for the ultimate reality. Our social roles, our possessions, and our ego-driven identity are all partof māyā. 
• The Persona (Jung): This is the “social mask” we wear to function in society. It’s our job title, our role as a parent or friend, and the personality we project. 
The Connection: Jung warned that identifying too much with the Persona (believing your mask is you) leads to spiritual and psychological emptiness. This is the same as the Upanishadic warning against being trapped by Māyā (mistaking the illusion for the reality) and forgetting your true nature as Ātman.
In short, Jung provided a modern, Western psychological framework to describe the same inner realities that the Upanishads explored thousands of years ago from a metaphysical and spiritual perspective.
For a deeper exploration of this, you might find this discussion on Jung’s connection to Eastern thought interesting: Carl Jung and The Eastern Philosophies. This video is relevant as it directly addresses how Jung’s psychological theories were influenced by and parallel the concepts found in Eastern spiritual traditions.