Task: Imaginary base numbers
C++: #include <algorithm>
#include <complex>
#include <iomanip>
#include <iostream>

std::complex<double> inv(const std::complex<double>& c) {
    double denom = c.real() * c.real() + c.imag() * c.imag();
    return std::complex<double>(c.real() / denom, -c.imag() / denom);
}

class QuaterImaginary {
public:
    QuaterImaginary(const std::string& s) : b2i(s) {
        static std::string base("0123.");

        if (b2i.empty()
            || std::any_of(s.cbegin(), s.cend(), [](char c) { return base.find(c) == std::string::npos; })
            || std::count(s.cbegin(), s.cend(), '.') > 1) {
            throw std::runtime_error("Invalid base 2i number");
        }
    }

    QuaterImaginary& operator=(const QuaterImaginary& q) {
        b2i = q.b2i;
        return *this;
    }

    std::complex<double> toComplex() const {
        int pointPos = b2i.find('.');
        int posLen = (pointPos != std::string::npos) ? pointPos : b2i.length();
        std::complex<double> sum(0.0, 0.0);
        std::complex<double> prod(1.0, 0.0);
        for (int j = 0; j < posLen; j++) {
            double k = (b2i[posLen - 1 - j] - '0');
            if (k > 0.0) {
                sum += prod * k;
            }
            prod *= twoI;
        }
        if (pointPos != -1) {
            prod = invTwoI;
            for (size_t j = posLen + 1; j < b2i.length(); j++) {
                double k = (b2i[j] - '0');
                if (k > 0.0) {
                    sum += prod * k;
                }
                prod *= invTwoI;
            }
        }

        return sum;
    }

    friend std::ostream& operator<<(std::ostream&, const QuaterImaginary&);

private:
    const std::complex<double> twoI{ 0.0, 2.0 };
    const std::complex<double> invTwoI = inv(twoI);

    std::string b2i;
};

std::ostream& operator<<(std::ostream& os, const QuaterImaginary& q) {
    return os << q.b2i;
}


QuaterImaginary toQuaterImaginary(const std::complex<double>& c) {
    if (c.real() == 0.0 && c.imag() == 0.0) return QuaterImaginary("0");

    int re = (int)c.real();
    int im = (int)c.imag();
    int fi = -1;
    std::stringstream ss;
    while (re != 0) {
        int rem = re % -4;
        re /= -4;
        if (rem < 0) {
            rem = 4 + rem;
            re++;
        }
        ss << rem << 0;
    }
    if (im != 0) {
        double f = (std::complex<double>(0.0, c.imag()) / std::complex<double>(0.0, 2.0)).real();
        im = (int)ceil(f);
        f = -4.0 * (f - im);
        size_t index = 1;
        while (im != 0) {
            int rem = im % -4;
            im /= -4;
            if (rem < 0) {
                rem = 4 + rem;
                im++;
            }
            if (index < ss.str().length()) {
                ss.str()[index] = (char)(rem + 48);
            } else {
                ss << 0 << rem;
            }
            index += 2;
        }
        fi = (int)f;
    }

    auto r = ss.str();
    std::reverse(r.begin(), r.end());
    ss.str("");
    ss.clear();
    ss << r;
    if (fi != -1) ss << '.' << fi;
    r = ss.str();
    r.erase(r.begin(), std::find_if(r.begin(), r.end(), [](char c) { return c != '0'; }));
    if (r[0] == '.')r = "0" + r;
    return QuaterImaginary(r);
}

int main() {
    using namespace std;

    for (int i = 1; i <= 16; i++) {
        complex<double> c1(i, 0);
        QuaterImaginary qi = toQuaterImaginary(c1);
        complex<double> c2 = qi.toComplex();
        cout << setw(8) << c1 << " -> " << setw(8) << qi << " -> " << setw(8) << c2 << "     ";
        c1 = -c1;
        qi = toQuaterImaginary(c1);
        c2 = qi.toComplex();
        cout << setw(8) << c1 << " -> " << setw(8) << qi << " -> " << setw(8) << c2 << endl;
    }
    cout << endl;

    for (int i = 1; i <= 16; i++) {
        complex<double> c1(0, i);
        QuaterImaginary qi = toQuaterImaginary(c1);
        complex<double> c2 = qi.toComplex();
        cout << setw(8) << c1 << " -> " << setw(8) << qi << " -> " << setw(8) << c2 << "     ";
        c1 = -c1;
        qi = toQuaterImaginary(c1);
        c2 = qi.toComplex();
        cout << setw(8) << c1 << " -> " << setw(8) << qi << " -> " << setw(8) << c2 << endl;
    }

    return 0;
}

Python: import math
import re

def inv(c):
    denom = c.real * c.real + c.imag * c.imag
    return complex(c.real / denom, -c.imag / denom)

class QuaterImaginary:
    twoI = complex(0, 2)
    invTwoI = inv(twoI)

    def __init__(self, str):
        if not re.match("^[0123.]+$", str) or str.count('.') > 1:
            raise Exception('Invalid base 2i number')
        self.b2i = str

    def toComplex(self):
        pointPos = self.b2i.find('.')
        posLen = len(self.b2i) if (pointPos < 0) else pointPos
        sum = complex(0, 0)
        prod = complex(1, 0)
        for j in xrange(0, posLen):
            k = int(self.b2i[posLen - 1 - j])
            if k > 0:
                sum += prod * k
            prod *= QuaterImaginary.twoI
        if pointPos != -1:
            prod = QuaterImaginary.invTwoI
            for j in xrange(posLen + 1, len(self.b2i)):
                k = int(self.b2i[j])
                if k > 0:
                    sum += prod * k
                prod *= QuaterImaginary.invTwoI
        return sum

    def __str__(self):
        return str(self.b2i)

def toQuaterImaginary(c):
    if c.real == 0.0 and c.imag == 0.0:
        return QuaterImaginary("0")

    re = int(c.real)
    im = int(c.imag)
    fi = -1
    ss = ""
    while re != 0:
        re, rem = divmod(re, -4)
        if rem < 0:
            rem += 4
            re += 1
        ss += str(rem) + '0'
    if im != 0:
        f = c.imag / 2
        im = int(math.ceil(f))
        f = -4 * (f - im)
        index = 1
        while im != 0:
            im, rem = divmod(im, -4)
            if rem < 0:
                rem += 4
                im += 1
            if index < len(ss):
                ss[index] = str(rem)
            else:
                ss += '0' + str(rem)
            index = index + 2
        fi = int(f)
    ss = ss[::-1]
    if fi != -1:
        ss += '.' + str(fi)
    ss = ss.lstrip('0')
    if ss[0] == '.':
        ss = '0' + ss
    return QuaterImaginary(ss)

for i in xrange(1,17):
    c1 = complex(i, 0)
    qi = toQuaterImaginary(c1)
    c2 = qi.toComplex()
    print "{0:8} -> {1:>8} -> {2:8}     ".format(c1, qi, c2),

    c1 = -c1
    qi = toQuaterImaginary(c1)
    c2 = qi.toComplex()
    print "{0:8} -> {1:>8} -> {2:8}".format(c1, qi, c2)
print

for i in xrange(1,17):
    c1 = complex(0, i)
    qi = toQuaterImaginary(c1)
    c2 = qi.toComplex()
    print "{0:8} -> {1:>8} -> {2:8}     ".format(c1, qi, c2),

    c1 = -c1
    qi = toQuaterImaginary(c1)
    c2 = qi.toComplex()
    print "{0:8} -> {1:>8} -> {2:8}".format(c1, qi, c2)

print "done"