Does Catholicism have a notion of "the God in me"?

The idea of God within can either be explained rightly, or really badly.

In the right understanding, Grace is the Holy Spirit entering us and infusing Divinity within us, making us a temple of the Spirit. Santifying Grace is partaking in Divinity, and can be understood as God placing Himself within us, to makes us Holy, for only God is Holy, so we must become God (by participation) in order to be Holy.

Moreover, I will give you a new heart and put a new spirit within you; and I will remove the heart of stone from your flesh and give you a heart of flesh. And I will put my Spirit in you and move you to follow my decrees and be careful to keep my laws. (Ezekiel 36: 26-27)

St. Catherine of Genoa also wrote a book on purgatory, in which she explains it as an inner fire which purifies us from within, which could be understood as santifying Grace, after being infused in our being, purifying us.

The difference between the Christian view and the Hindu view is that Hindus believe that humans are inherently Divine, and have just "forgotten" it. Christians, on the other hand, believe that humans are only Divine by Grace, a free gift given and not earned nor inherent in our nature, and primarily received in Baptism (where we are born again). In other words, Hindus reject the Creator/Creation divide, and advocates panentheism (in the mainstream understanding: there are many paths or yogas in Hinduism, many which lead to contradictory truths), while Christians are "quasi-panentheists."

To a Christian, only the Father (who is the source of Divintiy), the Son (who receives Divintiy through eternal begotteness), and the Holy Spirit (who receives Divinity through eternal "spiriting") are essentially Divine, and everyone else is Divine only accidently, only by Grace (we receive Divinity by participation/Grace/Divine Adoption). Hinduism claims everyone and everything is Divine essentially.

I find Hinduism to be one of the few religions that can actually rival Christianity, personally, in part due to how close its understanding is to the Christian understanding: a mixture of 90%Truth and 10%Lies is far more convincing than one of 10%Truth and 90%Lies.

Everything I wrote is based on multiple Fathers, St. Thomas Aquinas, and Father Reginald Garrigou-Lagrange. Here's a great quote from Father Garrigou-Lagrange (if you understand basic Aristotelian and Thomist philosophy and Western theological language):

On the other hand, sanctifying grace as such is not a participation in being in general, nor in life in general, nor in intellectuality in general, but a participation in Deity, which is found naturally only in God. Thus only grace is called a participation in the divine nature according as it is in us the radical principle of operations strictly divine, of which the formal object is (in heaven, at least) absolutely the same as the formal object of the uncreated operations of God.

All of this may be diagrammed as follows:

Thus the stone participates in being and has a likeness to God on the basis of being; grace, on the contrary, is directly and immediately a participation in the divine nature, not in any perfection analogically shared by God and the creature.

Therefore Deity as such cannot be partaken of except by some essentially supernatural gift. And, conversely, grace cannot be essentially supernatural unless it is a formal and physical participation in the divine nature as divine, that is, in the intimate life of God, or Deity as Deity, ordaining us to the knowledge of God as He Himself knows Himself immediately and to the love of God as He loves Himself.

Furthermore, sanctifying grace is a participation in Deity as it is in itself and not merely as it is known to us. For it is produced in our soul by an immediate infusion altogether independently of our knowledge of the Deity; and just as Deity as such is communicated to the Son by eternal generation, so Deity as such is partaken of by the just, especially by the blessed, through divine adoption.7

Hence, materially, grace is a finite accident, an entitative habit, but formally it is a formal participation in Deity as it is in itself, as it subsists in the three persons. Thus it is clearly evident that Deity as such in a certain sense surpasses being and intellection, since all absolutely simple perfections are identified in the eminence of Deity and can be naturally participated in, but Deity cannot be participated in naturally. (Cf. below, pp. 138 ff.: The dignity of sanctifying grace.)

Link: http://www.ewtn.com/library/Theology/grace3.htm

Christi pax.

Starting with Sacred Scripture, the old fancy Fr. Knox Bible available on New Advent says

the kingdom of God is here, within you. [1]

Luke 17:21

And that appears to agree with the notion that if you want to go looking for God you should start within yourself. But, lo, there is a footnote

[1] ‘Within you’; the Greek might also mean, ‘among you’.

And CCD folks have patched that a bit to make it clear so the New American Bible says

For behold, the kingdom of God is among you.

Luke 17:21 NABRE

And that makes sense, the "within you" translation seems to be acknowledged by those who wish to make something of it, but it makes no sense in conjunction with every other thing Our Lord or the Saints said about the Kingdom of God.

Other parts of the Bible imply that the Kingdom of God is so close you can taste It. But in modern English it's pretty important to distinguish between "within you" and "radiates from your pores" and it's even more important, given the secular Buddhism taught in public schools in the form of self-esteem promotion that we remember the very precise way Catholicism differs from Buddhism

The Catholic Church rejects nothing that is true and holy in these religions. She regards with sincere reverence those ways of conduct and of life, those precepts and teachings which, though differing in many aspects from the ones she holds and sets forth, nonetheless often reflect a ray of that Truth which enlightens all men. Indeed, she proclaims, and ever must proclaim Christ "the way, the truth, and the life" (John 14:6), in whom men may find the fullness of religious life, in whom God has reconciled all things to Himself.

http://www.vatican.va/archive/hist_councils/ii_vatican_council/documents/vat-ii_decl_19651028_nostra-aetate_en.html

Heresies always take something within Catholicism and run with it and heresies are rarely new. This is The Quaker Doctrine of the Inner Light and Tolstoyism which synthesizes the what is essentially the social teachings of the Church. I can't tell you exactly what is wrong with it, but I got a good sense when reading The Ball and the Cross

"But then he came," broke out MacIan, "and my soul said to me: 'Give up fighting, and you will become like That. Give up vows and dogmas, and fixed things, and you may grow like That. You may learn, also, that fog of false philosophy. You may grow fond of that mire of crawling, cowardly morals, and you may come to think a blow bad, because it hurts, and not because it humiliates. You may come to think murder wrong, because it is violent, and not because it is unjust. Oh, you blasphemer of the good, an hour ago I almost loved you! But do not fear for me now. I have heard the word Love pronounced in his intonation; and I know exactly what it means. On guard!'"

The Ball and the Cross - Chapter 5

So, I skipped about 1800 years of Catholic teaching on the matter, but my guess is that they only recently had to completely replace "within you" with "among you" or even "in your midst" (in the NIV) because people recently (with starting with Tolstoy?) developed a difficulty understanding something that plain.

Sister Simone kindly responded to my email asking this question. She said:

"If I just said Holy Spirit, it would slide over like water. Also to an interfaith audience who doesn't necessarily hold a Trinitarian faith it seemed more inclusive to me. I also know that God continues to create us at every moment so it isn't just the "image" of God but God who keeps us loved and alive."