Sometimes we know just enough about a vanished culture to pique our interest but not enough to form a clear picture. The result, if you're anything like me: a sort of pleasant frustration and a sense of the grand mystery and complexity of life. If you're nothing like me: congratulations! You're normal and not unbearably pretentious.

What happened: In the early twentieth century, researchers working in the Tarim Basin (western China) discovered texts in an unknown language. After sprinkling some linguistic magic around, it was discovered that the language was Indo-European and hitherto unknown. Based on a misidentification with another ancient people, the language was named Tocharian.

The Tocharians themselves lived in the Tarim Basin for quite a while; the famous Tarim mummies may have been Tocharian speakers. They converted to Buddhism early on and, with their central position in Asia, were influential in spreading the religion to China, among other things (including the pipa, one of the most recognizably Chinese musical instruments). They were also a fairly literate culture, as these things go; enough of their writings survive for us to construct a pretty good idea of their grammar and vocabulary—and to teach courses on it at the University of Texas, which, okay, was honestly the last thing I expected to find while researching. Their location on the Silk Road made them wealthy and cosmopolitan.

Alas, first the Chinese under the Tang Dynasty and then the Uyghur Turks moved into the region and displaced or absorbed the Tocharian inhabitants. Today, little remains of their culture beyond scraps of their literature and some frescoes.

What didn't happen: There's little historical reason to believe the Tocharians could have survived much longer, not with Tang China in the east and Genghis Khan in the future. But heck: let's say they did.

Tocharian forms its own branch of the Indo-European language tree, and a larger extant corpus of Tocharian literature would advance the study of the Proto-Indo-European people and their culture considerably. Imagine the huge and complicated mythology they would have recorded, imagine the poems and songs and stories, all in a unique, beautiful language. If you don't think this is a cool thing, I'm not sure we can be friends anymore.

An independent state of Tocharia might be carved out of the arid land between Russia and China. The city of Kucha would be roughly as large and important as it was in the early Common Era, and like many Silk Road cities it would be stunning. Tocharia would probably be conquered, like the rest of the world, by the Mongol Empire. However, we might be looking at a resurgence of Tocharian culture and national identity following the Empire's collapse.

Additionally, given the unique strains of Buddhism that developed in the various nations and empires of Asia, and the corresponding influence on the arts (Zen alone has been huge, as you might have noticed), I for one would love to see what Tocharian Buddhism and its attendant cultural manifestations would look like. 

The Tocharians or Tokharians were inhabitants of medieval oasis city-states on the northern edge of the Tarim Basin (modern Xinjiang, China). Their Tocharian languages (a branch of the Indo-European family) are known from manuscripts from the 6th to 8th centuries AD, after which they were supplanted by the Turkic languages of the Uyghur tribes.

Some scholars have linked the Tocharians with the Afanasevo culture of eastern Siberia (c. 3500 – 2500 BC), the Tarim mummies (c. 1800 BC) and the Yuezhi of Chinese records, most of whom migrated from western Gansu to Bactria in the 2nd century BC and then later to northwest India where they founded the Kushan Empire.

The Tocharians, living along the Silk Road, had contacts with the Chinese, the Persians, the Indians and Turkic tribes. They adopted Buddhism, which, like their alphabet, came from northern India in the 1st century of the 1st millennium, through the proselytism of Kushan monks. The Kushans and the Tocharians seem to have played a part in the Silk Road transmission of Buddhism to China. Many Tocharians apparently also practised some variant of Manichaeism.