Confusion while evaluating the complex continued fraction $\frac{1}{i+\frac{1}\ddots}$

Well, Vasily has put his finger on one problem, but I would like to point out a much more serious one.

We write a continued fraction to get a number that is the limit of the convergents, that is, of the expressions that you get when you cut your continued fraction off, to be a finite c.f.

The first convergent is $\frac1i$, no problem, but the second is $$\frac1{i+\frac1i}=\frac10\,,$$ a most unfortunate development. My recommendation would be to go on to other complex continued fractions, where the partial denominators are rather larger than $i$.

The problem with this expression is that function $f(x)=(i+x)^{-1}$ is a tricky one:

$$ f(f(x)) = \frac{1}{i+\frac{1}{i+x}}=\frac{i+x}{-1+ix+1}=-i+\frac1x,\\ f(f(f(x))) = \frac{1}{i+(-i+1/x)}=x. $$

So every number generates the orbit of length 3, except for two numbers you have found, which are fixed points. Thus, the series of nesting functions $f$ does not converge if you haven't already started with a fixed point. So the question as it is stated has no sense: you cannot assign any number to this expression.