#include <boost/multiprecision/cpp_complex.hpp>

namespace boost{ namespace multiprecision{

   template <unsigned Digits, backends::digit_base_type DigitBase = backends::digit_base_10, class Allocator = void, class Exponent = int, Exponent MinExponent = 0, Exponent MaxExponent = 0>
   using cpp_complex_backend = complex_adaptor<cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinExponent, MaxExponent> >;

   template <unsigned Digits, backends::digit_base_type DigitBase = digit_base_10, class Allocator = void, class Exponent = int, Exponent MinExponent = 0, Exponent MaxExponent = 0, expression_template_option ExpressionTemplates = et_off>
   using cpp_complex = number<complex_adaptor<cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinExponent, MaxExponent> >, ExpressionTemplates>;

   typedef cpp_complex<50> cpp_complex_50;
   typedef cpp_complex<100> cpp_complex_100;

   typedef cpp_complex<24, backends::digit_base_2, void, boost::int16_t, -126, 127> cpp_complex_single;
   typedef cpp_complex<53, backends::digit_base_2, void, boost::int16_t, -1022, 1023> cpp_complex_double;
   typedef cpp_complex<64, backends::digit_base_2, void, boost::int16_t, -16382, 16383> cpp_complex_extended;
   typedef cpp_complex<113, backends::digit_base_2, void, boost::int16_t, -16382, 16383> cpp_complex_quad;
   typedef cpp_complex<237, backends::digit_base_2, void, boost::int32_t, -262142, 262143> cpp_complex_oct;


}} // namespaces

The cpp_complex_backend back-end is used in conjunction with number: It acts as an entirely C++ (header only and dependency free) complex number type that is a drop-in replacement for std::complex, but with much greater precision.

The template alias cpp_complex avoids the need to use class number directly.

Type cpp_complex can be used at fixed precision by specifying a non-zero Digits template parameter. The typedefs cpp_complex_50 and cpp_complex_100 provide complex number types at 50 and 100 decimal digits precision respectively.

Optionally, you can specify whether the precision is specified in decimal digits or binary bits - for example to declare a cpp_complex with exactly the same precision as std::complex<double> one would use cpp_complex<53, digit_base_2>. The typedefs cpp_complex_single, cpp_complex_double, cpp_complex_quad, cpp_complex_oct and cpp_complex_double_extended provide software analogues of the IEEE single, double, quad and octuple float data types, plus the Intel-extended-double type respectively. Note that while these types are functionally equivalent to the native IEEE types, but they do not have the same size or bit-layout as true IEEE compatible types.

Normally cpp_complex allocates no memory: all of the space required for its digits are allocated directly within the class. As a result care should be taken not to use the class with too high a digit count as stack space requirements can grow out of control. If that represents a problem then providing an allocator as a template parameter causes cpp_complex to dynamically allocate the memory it needs: this significantly reduces the size of cpp_complex and increases the viable upper limit on the number of digits at the expense of performance. However, please bear in mind that arithmetic operations rapidly become very expensive as the digit count grows: the current implementation really isn't optimized or designed for large digit counts. Note that since the actual type of the objects allocated is completely opaque, the suggestion would be to use an allocator with char value_type, for example: cpp_complex<1000, digit_base_10, std::allocator<char> >.

The next template parameters determine the type and range of the exponent: parameter Exponent can be any signed integer type, but note that MinExponent and MaxExponent can not go right up to the limits of the Exponent type as there has to be a little extra headroom for internal calculations. You will get a compile time error if this is the case. In addition if MinExponent or MaxExponent are zero, then the library will choose suitable values that are as large as possible given the constraints of the type and need for extra headroom for internal calculations.

Finally, as with class number, the final template parameter determines whether expression templates are turn on or not. Since by default this type allocates no memory, expression template support is off by default. However, you should probably turn it on if you specify an allocator.

There is full standard library support available for this type, comparable with what std::complex provides.

Things you should know when using this type:

example:

#include <iostream>
#include <complex>
#include <boost/multiprecision/cpp_complex.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::cpp_complex_quad>();

    return 0;
}

Which produces the output (for the multiprecision type):

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)    : (0.696706709347165389063740022772448,0.717356090899522792567167815703377)

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) = 1.543080634815243778477905620757062
tan(z1) = (0.000000000000000000000000000000000,0.761594155955764888119458282604794)
asin(z1) = (0.000000000000000000000000000000000,0.881373587019543025232609324979793)
acos(z1) = (1.570796326794896619231321691639751,-0.881373587019543025232609324979793)
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)