The biggest barrier to approximating i, the imaginary number, as we approximate pi is the lack of a number -1. It is not clear how to take the square root of a negative symbol.
Interestingly, there are numeric representations of -1 in the left infinite numbers, the numbers that can continue to arbitrary large terms usually written to the left; …111. This has a similarity with real numbers that can continue to arbitrary small terms, usually written to the right; 1.111…
With these left infinite numbers, the left infinite string of (base -1) is a representation of -1. To see this, consider the example …999 . Adding …999+1 = …99(10) = …9(10)0 = …(10)00 = …000 which is zero. Base 10 can be replaced with any usual base. The import property is the carry.
This may seem strange, but this concept is used in most computers as the 2s complement representation. This representation uses all 1s to represent negative 1 and takes advantage of the natural arithmetic available.