A mere tool. That's what Hitori was. The accumulation of karma in Homura gave rise to another homura. A conscious Homura, with her own interests. Despite sharing a soul and appearance, this Homura felt different. Without past or future, but with the memories of the original. All the time travel she did, everything she sacrificed. This Homura was aware of this.
With a past of her own creation, knowing that the reason for her existence was to help Homura save Madoka, this Homura decided to be different. If she pretended to be someone else, Homura would trust her more. Despite being a mere temporary creation, a living paradox, she wanted to feel human. This is how she became known as Hitori. Born the same day Homura woke up in the hospital after another time trip, she was condemned to live a short life.
Hitori always knew that her existence would end early. The moment Homura defeated Walpurgis, at that point, Hitori would not be needed. A simple tool with a single objective, destined to be discarded once it has accomplished it.
However…she esteemed the original Homura. She knew that she wouldn't have that long of a life, but somehow or another, Homura earned a special place in her heart. Homura was the reason for her existence.
Hitori wanted to see what would happen if her soul gem became totally corrupted. So she fought with many witches without using the grief seeds they gave her. The moment her soul gem became corrupted, nothing happened. Hitori still existed.
"Why? "Why haven't I become a witch?"
It was then that Hitori understood. She had no soul. Born from Homura's suffering, she was just an empty shell. She is emptier than the magical girls themselves. Her and Homura's souls were the same. If Homura didn't become a witch, neither would Hitori, no matter how much Hitori's soul gem was corrupted.
"Why do I even exist…? Why can I think?! Why am I so human?!"
It was what Hitori asked herself every day. She wanted to live. She knew that her existence was paradoxical, but she wanted to continue existing. She couldn't live with the idea of ceasing to exist so suddenly, but her mere existence, the existence of two Homuras at the same time, was meaningless.
That's why Hitori decided to act so coldly towards Homura. With hers another me of hers. She couldn't let the original get attached to someone who would soon cease to exist. Hitori didn't want Homura to suffer any more than she already had. She didn't want to feel affection for Homura either. Why feel affection for others, if you know that your time is so short? Why would a tool have friends?
Getting up from her desk, Hitori formulated her goal: protect hers another me at all costs. The same way Homura cared for Madoka. If Madoka died, or something bad happened to her, Homura would travel back in time. She would accumulate more karma, which would discard Hitori in this line, making way for another new homura more capable of helping herself.
"If I am condemned to die, at least… at least I must help Homura. Whatever happens, she must be safe."