Star Child

"'So, you finally did make it. I've been waiting for you, you throne-bound, angry, incre-he-hedibly hypocritical man. I'm going to feel right at home.'"The Star Child is a mysterious entity who lives in the warp. He is a being of immense power that is connected to the Emperor of Mankind.

Official Canon
The Star Child was a theory put forward as far back as 1st Edition WH40K. The theory is that if the Emperor were to die, there is a chance that he will simply regenerate in the warp and be reborn, much like his Shaman ancestors had. Because much of his soul is fragmented across the warp and yet tied down to a dying physical body, being kept on the Golden Throne might be preventing his soul from healing. However, this is not a definitive conclusion.

TTS Canon
At some point prior or during the Emperor's rise to power he felt it necessary to remove his compassion, feeling that progress was more important. After his return to governing the Imperium he came to the conclusion that he will require his compassion once again and thus set in motion a plan to have it return to him.

This fragment, known as the Star Child, alongside Saint Celestine, are brought by the soul of Dominique to where Fyodor Karamazov and his remaining forces were found. The Star Child thanked Dominique for his work in preparing Karamazov, and set his plan into motion. The Star Child entered Karamazov’s body, filling him with divine power and transforming him into an image of the Emperor himself. Taken aback, Leman Russ recognized the transformed being as his father.

The Star Child uses his psychic powers to create a barrier to hold off the daemonic army as he and the Inquisition head through Khaine's Gate. After Draigo saved everyone from the sun that awaited them, the Star Child healed the Scions' injuries. The group heads to the entrance of Commoragh, where they are met by a host of Dark Eldar. Unlike the Inquisition and Russ, the Star Child insists on talking with the xenos instead of fighting them. The Star Child persuades his followers to peacefully follow the Dark Eldar.

The Star Child is brought to Asdrubael Vect's ziggurat. There he meets Vect and Urien Rakarth, Vect's Master Haemonculus. Vect knows the Star Child is the Emperor's exorcised compassion, merged with Fyodor. Urien explains that the Star Child's pure soul, when tortured, would ward Slaanesh from all Drukharii souls in Commoragh. The Star Child begins emitting an aura of "anti-negative" energy. This positive energy makes Vect happy and content with his life. As Urien tries various torture methods, this positive energy spreads throughout Commoragh, making the Dark Eldar empathetic and optimistic. Vect is on the verge of defeat when he reveals that he sent all of the Star Child's forces to fight in the Crucibael Arena. This knowledge distresses the Star Child enough to undo his positive energy and revive the Dark Eldar.

Trivia

 * In the Q&A session following the episode premiere, Alfabusa and Randolph explained that the Emperor deliberately used Karamazov as his soul's vessel. By taking the most stubborn man in the Imperium, convincing him that he is the Emperor, and sending him into the Warp, the nature of the Warp would naturally cause him to actually become the Emperor by guiding the Star Child to him.
 * Unlike the other fragments of the Emperor's soul the Star Child appears to be the only fragment he willingly separated from.