When Morgan Freeman pushed the 3 remaining bags of the CPH4 drug into Lucy, we saw her traveling through time (some steps of history from current world to big bang). Some of those events where witnessed by Lucy.
Did Lucy imagine those scenes or was she actually time-traveling? Because Lucy was back in the chair when Mr. Jang proceeded to shoot her. Did her mind experience the events or her mind just enacted those historical events as scientists give them by inference / resolution?
My take on it was that she could exist at all points in time. She transcended that as she had transcended other things in the movie. So she was actually moving through time, going wherever and whenever she wanted (including going through a black hole and to the "other side" where it connected with a quasar. At least I think that's what that was supposed to symbolize, and that visual has stuck with me). So she did in fact travel with her body and consciousness through time, and she did nearly touch the first protohuman Lucy's finger like God and Adam on the ceiling of the Sistine Chapel.
(I have no direct proof for this, like a quote from the writer or director. I'm basing it on everything that happened in the rest of the movie. She transcended the physical world, so next she'd transcend the temporal world.)
No, Lucy didn't travel in time (although, it was my first thought as well), nor did her mind re-enact those events. She remembered them.
Throughout the film, professor Norman explains how the purpose of life is to pass on the knowledge through time. From the first cell, its splitting in two, and on to the more complicated life forms, the purpose is always passing the information.
Also, among the first changes that Lucy experiences is her remembering her own early life (the cat that she had when she was one, her mother's milk,...) which she reveals in her phone conversation with her mother. Later on, that remembrance then spurs to the earlier events (her own growth of bones).
The "time travelling sequence" that we see is the process of all life since its beginning, in reverse. It does not end with the Big Bang. It ends with many cells connecting (back) until there is one left, which was the first cell ever. The whole sequence is Lucy remembering all the knowledge that was passed from that first cell all the way to her today's cells, and conveniently saving it in some form on a computer.
Addendum: It was noted in the comments that she had some interactions with history's beings (Indians, dinosaurs, human ancestors,...) and that we see her in those interactions, not her own ancestors. The way I see it is:
The memories are someone's. They acknowledged that person, not Lucy. We see her because she's remembering it.
I think it would've been too confusing if in each of those short interaction scenes we saw the actual ancestor, and the fact that no one seemed alarmed by her sudden appearance (which would be a necessity if she time-travelled) makes me more certain that those humans and animals interact with someone who was naturally there.
An illustrator from ILM who worked on the scene talked about his discussions with Besson where Lucy "would be travelling back through time until the moment of the Big Bang" in this article. So it appears that was the intent.
That being said, I think the genetic memories concept makes a lot more sense when you look at the actual images in the scene and all the apparent nods to that theory throughout the earlier film. The thing that seems to oppose the time travel idea by far I think is what was mentioned already about the last scene being the 1st cell splitting, not a big bang. Unless of course the idea was that the 1st cell dividing was actually the Big Bang, and our whole universe was created from that, or some other Men in Blacky type stuff. ;)
I don't think she travelled back in time. I understand that the Native Americans seemed to have seen her, and the dinosaur certainly did. But primate-Lucy, she didn't live in North America. Primate-Lucy lived in Ethiopia. If Omnipotent-Lucy time travelled, how is it possible she saw P-Lucy in to-be New York. Native Americans obviously lived in North America, and Lucy lived in Ethiopa. They could not have possibly lived in the same area at different times.
Omnipotent-Lucy had to have remembered them.
Reff: Lucy (2014) : Explained
I don't think she's traveling through time.
Once all the CPH4 goes in, her brain nears 90% utilization. As Lucy's brain function elevates further and further, she is able to tap into and use all forms of energy around her. She can look into the past or the stars and galaxy because they are all forms of light energy that she can access. (eg: The reflection of yourself in a mirror is you looking at the past because light takes finite time to travel from you to the mirror and back to your eyes)
She actually traveled in time - the Native Americans were able to look at her with a dazed expression.
