Without prior knowledge on the basics of complex numbers, this problem is impossible to solve. For example, it must be known that $i=\sqrt{-1}$ and that the rectangular coordinate form of a complex number is $z=a+bi$.

For the first subtask, since $x\le 1$, we can utilize a simple if statement. If $x=0$, the solution is $1+0i$. If $x=1$, the solution is simply $0+1i$.

For the second subtask, some quick Googling will give the solutions to when $x=2$ and when $x=3$. When $x=2$, the solution is $e^{-\pi /2}\approx 0.20788+0i$. When $x=3$, the solution is $e^{\pi i{e}^{-\pi /2}/2}\approx 0.94716+0.32076i$. Alternatively, one can solve for when $x=2$ and $x=3$ manually using Euler's formula.

For the third subtask, one can utilize a for loop that iterates from $1$ to $x$. Start with $2$ floating-point variables, $a_0$ set to $1$, and $b_0$ set to $0$. At each iteration of the for loop, it can be found that:

$${\displaystyle \begin{array}{rl}{a}_i& =e^{-\pi b_{i-1}/2}\cos \frac{\pi a_{i-1}}{2}\\ b_i& =e^{-\pi b_{i-1}/2}\sin \frac{\pi a_{i-1}}{2}\end{array}}$$

Alternatively, one can utilize a complex numbers library in their preferred programming language.

For the last subtask, one can figure out the solution converges to approximately $0.4383+0.3606i$, which means that after a certain $x$, the solution becomes consistent up to $4$ decimal places. This means that we can hardcode a value instead of looping up to $x$. This certain $x$ and the proof is left as an exercise for the reader.

Time Complexity: $\mathcal{O}(x)$