Purple in Roman History

The primary reason that the wearing of purple developed as a status symbol in ancient Rome is because it was expensive to import and therefore rare. Due to its rarity, it was later limited to the emporer and the upper class.

The whole history of clothing in ancient times is more complex than we see at face value.

Tyrian purple was reserved to emperors and royalty in later times not so much because it was difficult to import and thus expensive. It became a status symbol because of its price which was due to the fact that it took 12,000 snails to produce just 1.4 grams of this dye.

David Jacoby remarks that "twelve thousand snails of Murex brandaris yield no more than 1.4 g of pure dye, enough to colour only the trim of a single garment." - Wikipedia.

It was even too expensive for emperors at times!

Sometimes, however, the dye was too expensive even for royalty. Third-century Roman emperor Aurelian famously wouldn't allow his wife to buy a shawl made from Tyrian purple silk because it literally cost its weight in gold. Talk about sticker shock. - Why Is the Color Purple Associated With Royalty?

Tyrian Purple had some unique qualities:

It is said that it took 12,000 snails to produce just 1.4 grams of this dye. Because of this, it was so expensive, that the historian Theopompus reported that, "Purple for dyes fetched its weight in silver". Yet, there was a craze for this dye as a status symbol. In fact the Emperors of Byzantium made a law forbidding anybody from using it except themselves. The expression 'born in the purple' rose from this practice. In the picture, you can see the Emperor Justinian I dressed in a robe dyed with Tyrian Purple. Interestingly, unlike other dyes that faded in sunlight, Tyrian purple would become darker. - Tyrian Purple: the Colour of Kings.

Silk was also imported from China and India which meant that only the rich could afford it.

Silk and cotton were imported, from China and India respectively. Silk was rare and expensive; a luxury afforded only to the rich. Due to the cost of imported clothing, quality garments were also woven from nettle.1

Wild silk, that is, cocoons collected from the wild after the insect had eaten its way out, also was known.2 Wild silk, being of smaller lengths, had to be spun. A rare luxury cloth with a beautiful golden sheen, known as sea silk, was made from the long silky filaments or byssus produced by Pinna nobilis, a large Mediterranean seashell. - Clothing in ancient Rome.

In the East, saffron dyed clothing symbolized status.

At the court of the Sumerian king Gilgamesh only the court nobility wore saffron-dyed clothes. These clothes also belonged to the typical costume of Persian kings. - Saffron.

In China Yellow dyed fabrics were reserved to the Imperial Court:

In the Ming and Qing dynasties, the color yellow became exclusive to the imperial court. Civilians were prohibited from wearing yellow clothes. The clothing worn by emperors was called Yellow Robes. The carriage used by the emperor was called the Yellow House; the roads walked by the emperor were called the Yellow Path; the banners flown by the emperor during royal inspection tours were yellow. The seal of dynastic power was wrapped with yellow fabric. And only the imperial family could dwell in the special buildings built with red walls and yellow tiles. - Why Ancient Chinese Culture Was a World Filled with Yellow.

The Roman color purple might have been any shade of red or purple, or range of shades, as far as I know, unless an expert in Roman colors wants to elaborate. Of course anything colored with actual Tyrian Purple dye would have a known shade.

You should remember that during the Principate period the emperor was the real but unofficial absolute ruler of the Roman Republic and avoided rubbing his absolute power in the faces of the senators and aristocrats by having many different and separate powers, titles, offices, ranks, and honors granted him by the senators.

As a senator, and the Princeps Senatus, or "First Senator" (Macrinus in 217 AD was the first man to become emperor without becoming a senator first) the emperor would wear a normal senatorial toga with a broad red or purple stripe. Even the most important Romans only wore togas as formal ear, and wore tunics most of the time. Eventually the imperial tunic became the tunica palmata, all red or purple with embroidered golden palm leaves.

the "Severan Tondo" has a portrait of emperor Septimius Severus, his wife Julia Domna, and son Caracalla. They don't seem to be wearing any red or purple.

I am certain that a much better answer is possible.

There is an informative Wikipedia article.

A (reddish) purple dye extracted from Murex sea snails found on the coast of what is now Lebanon, ancient Phoenicia was a luxury product in the ancient Mediterranean world, being rare and therefore expensive and had the useful quality that instead of fading over time when worn in sunlight, it actually became brighter. Hence wearing purple was not only, arguably, an attractive colour it also demonstrated wealth.

As far as I know the Phoenician purple dye was not much exported as far as to China where a different tradition grew up that emperors wore yellow.