Date of Shankara the great saint of Hinduism is always been controversial. Dates ranging from 5th century BC to 9th century AD. Let us analyze the dates.
The most important internal evidence comes from Sankara's verbatim quotation of Dharmakirti, the buddhist logician. Hsuan Tsang , the Chinese pilgrim, who visited India in the time of Harshavardhana, king of Thanesar (606 - 647 CE), gives clues to Dharmakirti's date. He also mentions Bhartrhari , but not of Sankara. It follows that Sankara is post-Dharmakirti, and possibly post-Hsuan-Tsang also.
Various Sankara-vijaya texts are not of much use. The details in one work contradict those in another. Dates ranging from the 5th cent BCE to 8th cent CE have been calculated on the basis of such astronomical details. One further complication is that some astronomical information is said to have been obtained from works which are not available anywhere in India. So no firm conclusion can be drawn based on this.
Records of Mutts
Whether Sankara established any Mutts at all has been questioned. Paul Hacker attributes the tradition of four amnaya-maths at Sringeri, Puri, Dvaraka and Joshimath to Vidyaranyasvamin. The native oral tradition, takes the history of these four Mutts,each associated with one of the four geographical directions and one of the four vedas, to Sankaracarya himself. The dasanami sannyasi-sampradaya, with its various akhadas in northern India, accepts affiliation only with these four mutts. There historical evidence for the existence of the oldest dasanami akhadas as early as the 9th cent. CE. It is immaterial whether Sankara established them himself or whether these four mutts developed naturally at the places where the four famous disciples of Sankara lived and taught. Of these four mutts, the Joshimath title had long been vacant, till it was revived in 1940 CE. So, it does not have many ancient records. The Dvaraka and Puri mutts have, in the past, claimed a date of 5th century BCE for Sankara. This is partly based upon a dating of a grant by a king named Sudhanva who is supposed to have been a contemporary of Sankara. Nothing else is known about this king, and the grant itself has not been dated with any accuracy.And Dvaraka and puri mutts have patchy histories, with periods when there were no presiding Sankaracaryas.
Sringeri mutt in karnataka has been the only mutt of the original four which has had an unbroken succession of mathadhipatis, as southern India has not experienced as many political upheavals as the north. The Sringeri mutts record states that Sankara was born in the 14th year of the reign of Vikramaditya. The record does not give any clue about the identity of this king. Some 19th century researchers identified this king with the famous Vikramaditya of the Gupta dynasty, thereby postulating a date of 44 BCE for Sankara. Gap of 700 years was then assigned to Suresvara. The later successors in the Sringeri list can all be dated reasonably accurately from the 8th century downwards. If one identifies the Vikramaditya as a member of the Western Chalukya dynasty, which ruled from Badami in Karnataka. The Chalukya dynasty reached its greatest fame in the time of Pulakesin II, who defeated Harshavardhana. There were two kings named Vikramaditya in this Chalukya dynasty - Vikramaditya I ruled in the late 7th century CE, while Vikramaditya II ruled in the early 8th century. It is more reasonable to identify the Vikramaditya of the Sringeri record with one of these two Chalukyan kings, who ruled from Karnataka, rather than the northern gupta king, whose empire did not include southern India. This interpretation of the Sringeri record is also consistent with the internal evidence from Sankara's works.
In addition to these four original mutts, a number of other advaita mutts have come into being over the centuries, some of which are quite well-known. These mutts either started out as branches of the original institutions, or were set up as independent monasteries by notable sannyasis of the dasa-nami order. With the proliferation of such mutts came a number of "traditions," many of them conflicting with one another in details. some of these mutts also claim to have been established by Sankara himself., Some of them also claim 5th century BCE to be the date of Sankara.
The intimate connection of the founders of the Vijayanagara empire with the Sringeri mutt, and the competition by other mathadhipatis in the south for similar honors as traditionally accorded to the Sringeri Mutt. Every southern Mutt with a claim to be the "original" one wants to deny Sringeri's chronological primacy. This denial only has the effect of reinforcing the fact that Sringeri has been the most important advaita mutt for centuries. As such, their conflicting claims about Sankara's date have to be evaluated in the context of their political motivations in putting forth such dates.
Fifth advaita mutt at Kanchipuram is very active today, does not mean that it has always been so, nor does such activity lend any special credibility to its claims to antiquity. The political influence and prestige that a Mutt enjoys today also do not confer any legitimacy to such claims. It is inconceivable that the dasa-nami-sampradaya would have overlooked a fifth mutt in choosing its affiliations. Claims to historicity that are made in a spirit of political one-upmanship seldom stand up to serious scrutiny. There is no necessary correlation between the modern activity of an advaita mutt and its claimed antiquity.
The official date accepted currently is 788-820 CE, and the Government of India celebrated the 1200th anniversary of Sankara's birth in 1988. This date is largely based upon one traditional view prevalent in India.