This is an example of lingo
lingo : b : the special vocabulary of a particular field of interest
Here "code complete" indicates a stage or milestone in the software development lifecycle.
As is the case with all specialized vocabulary it should be used only with an audience who is expected to be able to understand it.
In this particular case, the usage is quite informal and to maintain the same register I'd suggest using contractions and I'd probably still use scare quotes around it:
I'm already working on it and it's almost "code complete".
Otherwise in a more formal setting, something like:
This work has already been started and coding is nearly complete.
might be better.