The answer to this question depends very greatly on what's meant by "comprehensive" and what's meant by "companies".

If "comprehensive" is taken to mean a detailed exploration of how and in what circumstances such groups came into existence, and then operated on a day-to-day basis, that would imply the survival of a fairly extensive body of representative records. And if "company" is taken to mean a group of people who come together in a formal organisation of some sort to jointly pursue a common commercial aim, and in which there is a division between shareholders, paid dividends, and employees, paid salaries, that would exclude some of the older commercial groupings for which records do survive, such as guilds – which assembled groups of people who were in the same trade, but were to at least some extent competitors with each other, in an organisation that sought more to protect and extend privileges than undertake joint commercial projects.

I'm not sufficiently expert in the business history of India or the Far East to say whether or not such records and such organisations do exist for ancient periods – hopefully other flairs will be able to help out in that regard. But certainly in Europe, companies in the modern sense of the word – which I'd take to mean organisations with a central pool of funds, supplied by shareholders, an ongoing business that went beyond single projects, and a hierarchy of employees, rather than, say, a group of independent merchants pooling their resources to achieve a single goal – did not exist until the early modern period.

The earliest examples are the large trading corporations formed to exploit the great opportunities (and share the great risks) that opened up in overseas trade as a result of the "Age of Exploration". The Dutch East India Company (VOC), which was founded in 1602 from the amalgamation of several earlier and smaller competing town-based trading groups, was the largest and most powerful of these. Like its counterpart the English East India Company, it benefitted from a cosy relationship with the state, which gave it a charter that guaranteed a monopoly on the importation of specified categories of goods, such as spices and dyes, from a specified area, which in the VOC's case stretched from Persia to Japan, in exchange for substantial regular payments made for renewals of that charter. This proved to be a very successful business model, and soon there were French, Danish and Swedish East India Companies as well. Chartered companies typically enjoyed the right to raise taxes in the territories they conquered, and to create private armies and navies to acquire land and trading rights and protect them against both local rulers and each other.

Such groups did have precursors, though, that might meet your own definitions and interest you. The East India companies were preceded by groups such as the Society of Merchant Venturers, a merchant group based in the port of Bristol, in the west of England, which grew out of the guild system and funded the voyages undertaken by John Cabot that resulted in the exploration of Newfoundland in the late 15th century. Going further back, the Hanseatic League, a defensive alliance of merchant guilds mainly based in German-speaking towns along the Baltic and North Sea coasts and dating to the late twelfth century, also had several of the characteristics of the later joint stock companies trading with the east.

One other comparison worth drawing is with the multinational business built up by Francesco Di Marco Datini, a highly successful merchant based in the northern Italian city of Prato during the 14th century. Although certainly not a company in the modern meaning of the term, Datini's business did have numerous employees and maintained a number of offices in other territories in southern Europe. It's worth mentioning in the context of your question because the almost complete records of the business were discovered in the mid nineteenth century lying in a forgotten under-stairs cupboard in his former home. They constitute the largest and most complete business archive that we have from the medieval period.

Some reading

G.L. Balk et al, The Archives of the Dutch East India Company (VOC) and the Local Institutions in Batavia (Jakarta) (2007)

E.M. Carus-Wilson (ed.), Medieval Merchant Venturers: Collected Studies (2006)

Philippe Dollinger, The German Hansa (1970)

Iris Origo, The Merchant of Prato: Daily Life in a Medieval Italian City (1992)

Tirthankar Roy, The East India Company: The World’s Most Powerful Corporation (2016)