Sankara's buddhism assimilation is extensively studied already. He is even addressed as 'prachhanna buddha' . Nevertheless I will list whatever I have found - just for my own satisfaction. Christian assimilation will be emphasised, though.
Thanks to Sanjay Chandrasekhararan for the information and pointers on the philosophical aspects of buddhism and advaita.
Please use http://www.wikipedia.org, for explanation of the keywords.
Buddha was born around 563 BC
Christ was born around 5 BC
Adi Sankara was born in 788 AD in Kaladi in the coast of Periyar.
Keralan Christianity at 800 AD
The coast of Periyar was not just any place in Tamil Kerala. Muzaris (Kodungaloor) - most important Indian port of that time - was at the mouth of Periyar. It was a melting pot of many cultures and religions. Chinese, Arabs, Romans, Greeks all had very active spice trade with India through that port. Thus that area came into contact with all great religions of the world.
Due to lack of historical evidences, I do not believe St. Thomas came to Kerala. May be his close disciples would have. I count Thomas Knana's arrival (>325 AD) as first (only?) major Christian migration to India. Before that whatever Christianity existed in Kerala would have been something like Jews in Kochi now :) Remember first major Brahmin migrations to Kerala also happened at that period ~300 AD.
Therefore, Christianity has nearly completed 500 years when Sankara was born. From Malayatoor to Angamali to Kodungaloor, there were many important Christian centers in the coast of Periyar. Remember those were the golden ages of Byzantian Christian Empire. Christian theology, traditions and images were all dominated by this powerful empire. Images were a lot sacred to them and Christians still use those famous icons. (refer Iconoclasm & and its eventual failure)
Thomas has a gospel of his own. It was discovered in 1945 from Nag Hammadi in Egypt. It is considered Gnostic. That means, it is drastically different text and forwards a different theology compared to other 4 gospels. St. Thomas' Gnostic theology and his gospel could have been, at the least, known among Christian intellectuals in the Periyar coast. As you may already know, Gnostism goes well with Buddhism & Advaita.
My view of Buddhism
In it essence, Buddhism is exploring the spirituality; and only spirituality. It is not interested in a God story. In fact, one can plug his own God story to Buddha's spiritual practices with little imagination. Buddha's conscious silence about his own God, I believe, is from his principle on never to let his disciples follow anything which he didn't experiment or test - as a scientist. Nevertheless, Buddhism couldn't satisfy an Indian's thirst for a God story & theory.
People who meditate are known to experience a universal oneness between self and other creatures and everything in the universe. This gives birth to Buddha's compassion to all. So a Buddhist's personal god theory can very well be Advaita.
What Sankara achieved was to successfully connect his Advaita experience to Vedas and Vedanthas and bring forward a God theory consistent with them.
Now to the evidences...
Look at these iconic images - Adi Sankara's sitting pose and Byzantine Christ (google image search will suffice). I would argue Sankara's image is a merge of Christian icons and his own Brahmanic & Shivite traditions. That is, his graphic designer could have been his Nasrani friend in Angamali :-)
- Hand posture is same as that of Jesus. The typical posture of a Hindu deity is different - it is full palm facing the observer or holding something. Hand postures are important because that signifies what he claims to be. Jesus' icons adopted Apollo's finger expression to claim that he is God. Sankara's posture may be presenting him as a great teacher like Jesus.
- The bible is replaced with Veda.
- The stick in his hand graphically resembles the cross
- The headscarf is still used by orthodox Christian bishops. I haven't seen any other Indian sects use headscarf this way.
Monistic quotes from Gospel of Thomas
- "the kingdom is within you, and it is outside of you. When you know yourselves, then you will be known, and you will know that you are the sons of the living Father" Quote 3
- "for there are many first who shall be last, and they will become a single one" Quote 4
- "When you make the two one and make the inside like the outside and the outside like the inside and the above like the below... - then you will enter the kingdom" Quote 22
- Jesus said: "If the flesh came into existence because of the spirit, it is a marvel. But if the spirit (came into existence) because of the body, it is a marvel of marvels." Quote 29
- Jesus said, "If they say to you (plur.), 'Where are you from?' say to them, 'It is from light that we have come - from the place where light, of its own accord alone, came into existence and [stood at rest]. And it has been shown forth in their image.' Quote 50
- Jesus said: I am the light that is above them all. I am the all; the all came forth from me, and the all attained to me. Cleave a (piece of) wood; I am there. Raise up a stone, and you will find me there. Quote 77
- Jesus said, "Images are visible to human beings. And the light within these (images) is hidden by the image of the father's light; it will be disclosed. And his image is hidden by his light." Quote 83
- "When you make the two into one, you will become like me (Son of Man)" Quote 106. "onnaaya ninneyiha randennu..." is the most popular quote from Advaita. (I still can not find the source of it)
- Jesus said: He who drinks from my mouth will become like me, and I will become like him, and the hidden things will be revealed to him. Quote 108
- Famous Snake-Rope Buddhist metaphor appears in Viveka Chudamani sloka 11.
Social Structure Evidence
- Buddhism & Christianity are great missionary religions. Hinduism never was. Sankara was the first Hindu missionary.
- Establishment of 7 & 1/2 churches of St. Thomas and 4 & 1/2 Mutts by Adi Sankara
- Unmarried monks staying together was not heard in Hinduism until Sankara. It has been practiced widely in Buddhism and Christianity.
This don't conclusively prove anything. This is just a thought line for new explorations to this observation.
There are more 'thathwamasi' in Gospel of Thomas and others. They needs to be listed. 'aham brahmasmi' in Gospels is left as an excersise to readers :)
Regarding the icons we will need to find the period of their origin. More than that, we need to investigate more into the 9th century of Kerala. The events in that century could have been the precursor to the turbulent period that followed. We need to ask what changed in the 8th and 9th century Keralan society and why.