There's no rules about renaming or not your company; it's your logo and your company, you do what you want. After all, a lot of people name their businesses based on symbols they like.
If this graphic has a special meaning to you, it's alright to use it. Especially in design, there's no limits to express your own style and since it's for yourself, you truly have no limits in what you want to do with that design either. Dali loved rhinoceros horns and ants... it didn't make sense to 99.999% of the planet and it didn't matter; he wasn't selling cars or shoes, he was selling his art. Go for what you like if you do something for yourself. E or C, it doesn't matter.
Lot of designers create logos with initials or letters only. Your graphic is just another way of showing a letter. The only problem I can see you may have with it is the RGB colors that will look a bit less bright when you'll convert it to a CMYK version for printing (if you ever need printed material.)
Just a side note, I think the colored part looks a bit like a bird.
Just throwing suggestions... if it can help
If you want to slightly modify your logo to a C, and don't mind losing that bit in the middle, there isn't so much work to do to modify it.
I don't know if this is what you think would look more like a C but here's a sample of your image modified. I only filled that area with the same black as your background, and used the brush tool at 105pt with no hardness for the part closer to the colored area.
Here's another variation if you want to accentuate the C (all RGB color):
I can see how this graphic could make a nice logo if you keep it in a black square box and write your company name below or on the right with a thick sans-serif in black. Maybe that's not a style you like, maybe it is.