The following mapping works fine here:
inoremap <up> <c-o>gk
Note that using Ctrl+O you don't need the
a at the end of your mapping to get back to insert mode. And you should generally avoid using using Ctrl+C to exit insert mode, as it can break some functionality and plugins.
You could try the following to debug your problem:
1) Check if the problem happens in vanilla vim
Start Vim with
vim -u NONE -U NONE -N -i NONE. If the problem doesn't happens, it means you have a problem related to some setting/plugin. In this case you could follow the steps at Vim-FAQ 2.5 - I have a "xyz" (some) problem with Vim. How do I determine it is a problem with my setup or with Vim? / Have I found a bug in Vim?.
2) Try to map another key instead of the arrow keys
Still with vanilla Vim, try mapping to a regular key:
inoremap a <c-o>gk
If the problem doesn't happens, check the instructions at Vim-FAQ 10.9 - When I use my arrow keys, Vim changes modes, inserts weird characters in my document but doesn't move the cursor properly. What's going on?, which references some relevant options such as
3) Try other terminals/gVim
If nothing else solves the problem, the problem is probably related to your terminal. You could try using another terminal or, even better, gVim (Graphical Vim), which is available on Windows and thus would allow you to avoid Cygwin and any problem related to shell limitations.