There may be no symbolic solution possible. Additionally, are you sure that this converges?
Products generally are hard and a common first step when working with them is to use Log to convert it a sum and then simplify. Doing this in your case doesn't make a symbolic solution apparent, but does provide a much more numerically stable way of calculating the result.
There is some telescoping that goes in in these terms: (-1 + 2 i)^(-(1/(-1 + 2 i))) (1 + 2 i)^(-(1/(1 + 2 i))). Notice that it uses -1+2i and 1+2i and so they can be replaced with ((-1 + 2 i)^(-(1/(-1 + 2 i))))^2
Putting these together, I can get your sum simplified to this, but haven't made any progress further:
Exp@NSum[(Log[2] + Log[i])/i + 2 Log [(-1 + 2 i)]/(1 - 2 i), {i, 1, Infinity}]
