#include <boost/multiprecision/complex128.hpp>

namespace boost{ namespace multiprecision{

class complex128_backend;

typedef number<complex128_backend, et_off>    complex128;

}} // namespaces

The complex128 number type is a very thin wrapper around GCC's __float128 or Intel's _Quad data types and provides a complex-number type that is a drop-in replacement for the native C++ floating-point types, but with a 113 bit mantissa, and compatible with FORTRAN's 128-bit QUAD real.

All the usual standard library functions are available, performance should be equivalent to the underlying native types.

As well as the usual conversions from arithmetic and string types, instances of float128 are copy constructible and assignable from GCC's __float128 and Intel's _Quad data types.

Things you should know when using this type:

complex128 example:

#include <iostream>
#include <complex>
#include <boost/multiprecision/complex128.hpp>

template<class Complex>
void complex_number_examples()
{
    Complex z1{0, 1};
    std::cout << std::setprecision(std::numeric_limits<typename Complex::value_type>::digits10);
    std::cout << std::scientific << std::fixed;
    std::cout << "Print a complex number: " << z1 << std::endl;
    std::cout << "Square it             : " << z1*z1 << std::endl;
    std::cout << "Real part             : " << z1.real() << " = " << real(z1) << std::endl;
    std::cout << "Imaginary part        : " << z1.imag() << " = " << imag(z1) << std::endl;
    using std::abs;
    std::cout << "Absolute value        : " << abs(z1) << std::endl;
    std::cout << "Argument              : " << arg(z1) << std::endl;
    std::cout << "Norm                  : " << norm(z1) << std::endl;
    std::cout << "Complex conjugate     : " << conj(z1) << std::endl;
    std::cout << "Projection onto Riemann sphere: " <<  proj(z1) << std::endl;
    typename Complex::value_type r = 1;
    typename Complex::value_type theta = 0.8;
    using std::polar;
    std::cout << "Polar coordinates (phase = 0)    : " << polar(r) << std::endl;
    std::cout << "Polar coordinates (phase !=0)    : " << polar(r, theta) << std::endl;

    std::cout << "\nElementary special functions:\n";
    using std::exp;
    std::cout << "exp(z1) = " << exp(z1) << std::endl;
    using std::log;
    std::cout << "log(z1) = " << log(z1) << std::endl;
    using std::log10;
    std::cout << "log10(z1) = " << log10(z1) << std::endl;
    using std::pow;
    std::cout << "pow(z1, z1) = " << pow(z1, z1) << std::endl;
    using std::sqrt;
    std::cout << "Take its square root  : " << sqrt(z1) << std::endl;
    using std::sin;
    std::cout << "sin(z1) = " << sin(z1) << std::endl;
    using std::cos;
    std::cout << "cos(z1) = " << cos(z1) << std::endl;
    using std::tan;
    std::cout << "tan(z1) = " << tan(z1) << std::endl;
    using std::asin;
    std::cout << "asin(z1) = " << asin(z1) << std::endl;
    using std::acos;
    std::cout << "acos(z1) = " << acos(z1) << std::endl;
    using std::atan;
    std::cout << "atan(z1) = " << atan(z1) << std::endl;
    using std::sinh;
    std::cout << "sinh(z1) = " << sinh(z1) << std::endl;
    using std::cosh;
    std::cout << "cosh(z1) = " << cosh(z1) << std::endl;
    using std::tanh;
    std::cout << "tanh(z1) = " << tanh(z1) << std::endl;
    using std::asinh;
    std::cout << "asinh(z1) = " << asinh(z1) << std::endl;
    using std::acosh;
    std::cout << "acosh(z1) = " << acosh(z1) << std::endl;
    using std::atanh;
    std::cout << "atanh(z1) = " << atanh(z1) << std::endl;
}

int main()
{
    std::cout << "First, some operations we usually perform with std::complex:\n";
    complex_number_examples<std::complex<double>>();
    std::cout << "\nNow the same operations performed using quad precision complex numbers:\n";
    complex_number_examples<boost::multiprecision::complex128>();

    return 0;
}

Which results in the output:

Print a complex number: (0.000000000000000000000000000000000,1.000000000000000000000000000000000)
Square it             : -1.000000000000000000000000000000000
Real part             : 0.000000000000000000000000000000000 = 0.000000000000000000000000000000000
Imaginary part        : 1.000000000000000000000000000000000 = 1.000000000000000000000000000000000
Absolute value        : 1.000000000000000000000000000000000
Argument              : 1.570796326794896619231321691639751
Norm                  : 1.000000000000000000000000000000000
Complex conjugate     : (0.000000000000000000000000000000000,-1.000000000000000000000000000000000)
Projection onto Riemann sphere: (0.000000000000000000000000000000000,1.000000000000000000000000000000000)
Polar coordinates (phase = 0)    : 1.000000000000000000000000000000000
Polar coordinates (phase !=0)    : (0.696706709347165389063740022772449,0.717356090899522792567167815703377)

Elementary special functions:
exp(z1) = (0.540302305868139717400936607442977,0.841470984807896506652502321630299)
log(z1) = (0.000000000000000000000000000000000,1.570796326794896619231321691639751)
log10(z1) = (0.000000000000000000000000000000000,0.682188176920920673742891812715678)
pow(z1, z1) = 0.207879576350761908546955619834979
Take its square root  : (0.707106781186547524400844362104849,0.707106781186547524400844362104849)
sin(z1) = (0.000000000000000000000000000000000,1.175201193643801456882381850595601)
cos(z1) = 1.543080634815243778477905620757061
tan(z1) = (0.000000000000000000000000000000000,0.761594155955764888119458282604794)
asin(z1) = (0.000000000000000000000000000000000,0.881373587019543025232609324979792)
acos(z1) = (1.570796326794896619231321691639751,-0.881373587019543025232609324979792)
atan(z1) = (0.000000000000000000000000000000000,inf)
sinh(z1) = (0.000000000000000000000000000000000,0.841470984807896506652502321630299)
cosh(z1) = 0.540302305868139717400936607442977
tanh(z1) = (0.000000000000000000000000000000000,1.557407724654902230506974807458360)
asinh(z1) = (0.000000000000000000000000000000000,1.570796326794896619231321691639751)
acosh(z1) = (0.881373587019543025232609324979792,1.570796326794896619231321691639751)
atanh(z1) = (0.000000000000000000000000000000000,0.785398163397448309615660845819876)