diff --git a/include/boost/multiprecision/float128.hpp b/include/boost/multiprecision/float128.hpp
index 81d7be741..378e2ced2 100644
--- a/include/boost/multiprecision/float128.hpp
+++ b/include/boost/multiprecision/float128.hpp
@@ -20,6 +20,7 @@
 #include <cfloat>
 #include <tuple>
 #include <cstring>
+#include <complex>
 #include <boost/multiprecision/detail/standalone_config.hpp>
 #include <boost/multiprecision/number.hpp>
 #include <boost/multiprecision/detail/hash.hpp>
@@ -710,6 +711,29 @@ inline boost::multiprecision::number<float128_backend, ExpressionTemplates> rsqr
    return res;
 }
 
+// The default std::abs(std::complex<>) implementation normalizes by max(|re|, |im|),
+// which yields NaN when an input is infinite (inf/inf).
+// Per IEEE 754, the result must be +infinity if either component is infinite, even if the other is NaN.
+template <boost::multiprecision::expression_template_option ExpressionTemplates>
+inline boost::multiprecision::number<float128_backend, ExpressionTemplates>
+abs BOOST_PREVENT_MACRO_SUBSTITUTION(const std::complex<boost::multiprecision::number<float128_backend, ExpressionTemplates>>& z)
+{
+   using number_type = boost::multiprecision::number<float128_backend, ExpressionTemplates>;
+   const float128_type re_v = z.real().backend().value();
+   const float128_type im_v = z.imag().backend().value();
+#ifdef BOOST_MP_USE_FLOAT128
+   return number_type(::hypotq(re_v, im_v));
+#else
+   if (isinfq(re_v) || isinfq(im_v))
+   {
+      return std::numeric_limits<number_type>::infinity();
+   }
+   const float128_type re_abs = re_v < 0 ? -re_v : re_v;
+   const float128_type im_abs = im_v < 0 ? -im_v : im_v;
+   return number_type(sqrtq(re_abs * re_abs + im_abs * im_abs));
+#endif
+}
+
 #ifndef BOOST_MP_USE_QUAD
 template <multiprecision::expression_template_option ExpressionTemplates>
 inline boost::multiprecision::number<boost::multiprecision::backends::float128_backend, ExpressionTemplates> copysign BOOST_PREVENT_MACRO_SUBSTITUTION(const boost::multiprecision::number<boost::multiprecision::backends::float128_backend, ExpressionTemplates>& a, const boost::multiprecision::number<boost::multiprecision::backends::float128_backend, ExpressionTemplates>& b)
diff --git a/test/Jamfile.v2 b/test/Jamfile.v2
index d261c6f58..ba8eb4c14 100644
--- a/test/Jamfile.v2
+++ b/test/Jamfile.v2
@@ -1281,6 +1281,7 @@ test-suite misc :
       [ run git_issue_636.cpp : : : [ check-target-builds ../config//has_float128 : <source>quadmath : <build>no ] ]
       [ run git_issue_652.cpp ]
       [ run git_issue_734.cpp ]
+      [ run git_issue_759.cpp : : : [ check-target-builds ../config//has_float128 : <source>quadmath : <build>no ] ]
       [ compile git_issue_98.cpp :
          [ check-target-builds ../config//has_float128 : <define>TEST_FLOAT128 <source>quadmath : ]
          [ check-target-builds ../config//has_gmp : <define>TEST_GMP <source>gmp : ]
diff --git a/test/git_issue_759.cpp b/test/git_issue_759.cpp
new file mode 100644
index 000000000..cd94a5a65
--- /dev/null
+++ b/test/git_issue_759.cpp
@@ -0,0 +1,68 @@
+///////////////////////////////////////////////////////////////////////////////
+//  Copyright 2026 Matt Borland. Distributed under the Boost
+//  Software License, Version 1.0. (See accompanying file
+//  LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
+//
+// See: https://github.com/boostorg/multiprecision/issues/759
+
+#include <complex>
+#include <iostream>
+#include <limits>
+#include <boost/multiprecision/float128.hpp>
+#include "test.hpp"
+
+int main()
+{
+   using REAL = boost::multiprecision::float128;
+
+   const REAL inf_v = std::numeric_limits<REAL>::infinity();
+
+   {
+      using std::abs;
+      const REAL r = abs(inf_v);
+      BOOST_CHECK((boost::multiprecision::isinf)(r));
+   }
+
+   {
+      using std::abs;
+      const std::complex<REAL> z{inf_v, REAL(0)};
+      const REAL r = abs(z);
+      BOOST_CHECK((boost::multiprecision::isinf)(r));
+      BOOST_CHECK(!(boost::multiprecision::isnan)(r));
+   }
+
+   {
+      using std::abs;
+      const std::complex<REAL> z{REAL(0), inf_v};
+      const REAL r = abs(z);
+      BOOST_CHECK((boost::multiprecision::isinf)(r));
+      BOOST_CHECK(!(boost::multiprecision::isnan)(r));
+   }
+
+   {
+      using std::abs;
+      const std::complex<REAL> z{-inf_v, REAL(3)};
+      const REAL r = abs(z);
+      BOOST_CHECK((boost::multiprecision::isinf)(r));
+      BOOST_CHECK(!(boost::multiprecision::isnan)(r));
+   }
+
+   {
+      // Per IEEE 754: hypot(inf, NaN) = inf.
+      using std::abs;
+      const REAL nan_v = std::numeric_limits<REAL>::quiet_NaN();
+      const std::complex<REAL> z{inf_v, nan_v};
+      const REAL r = abs(z);
+      BOOST_CHECK((boost::multiprecision::isinf)(r));
+   }
+
+   {
+      // Sanity check on a finite value: abs(3 + 4i) == 5.
+      using std::abs;
+      const std::complex<REAL> z{REAL(3), REAL(4)};
+      const REAL r = abs(z);
+      BOOST_CHECK_EQUAL(r, REAL(5));
+   }
+
+   return boost::report_errors();
+}