diff --git a/Lib/test/test_complex.py b/Lib/test/test_complex.py
--- a/Lib/test/test_complex.py
+++ b/Lib/test/test_complex.py
@@ -27,7 +27,7 @@ class ComplexTest(unittest.TestCase):
                 unittest.TestCase.assertAlmostEqual(self, a, b)
 
     def assertCloseAbs(self, x, y, eps=1e-9):
-        """Return true iff floats x and y "are close\""""
+        """Return true iff floats x and y "are close"."""
         # put the one with larger magnitude second
         if abs(x) > abs(y):
             x, y = y, x
@@ -62,7 +62,7 @@ class ComplexTest(unittest.TestCase):
         self.fail(msg.format(x, y))
 
     def assertClose(self, x, y, eps=1e-9):
-        """Return true iff complexes x and y "are close\""""
+        """Return true iff complexes x and y "are close"."""
         self.assertCloseAbs(x.real, y.real, eps)
         self.assertCloseAbs(x.imag, y.imag, eps)
 
@@ -104,6 +104,11 @@ class ComplexTest(unittest.TestCase):
         self.assertAlmostEqual(complex.__truediv__(2+0j, 1+1j), 1-1j)
         self.assertRaises(ZeroDivisionError, complex.__truediv__, 1+1j, 0+0j)
 
+        for denom_real, denom_imag in [(0, NAN), (NAN, 0), (NAN, NAN)]:
+            z = complex(0, 0) / complex(denom_real, denom_imag)
+            self.assertTrue(isnan(z.real))
+            self.assertTrue(isnan(z.imag))
+
     def test_floordiv(self):
         self.assertRaises(TypeError, complex.__floordiv__, 3+0j, 1.5+0j)
         self.assertRaises(TypeError, complex.__floordiv__, 3+0j, 0+0j)
diff --git a/Objects/complexobject.c b/Objects/complexobject.c
--- a/Objects/complexobject.c
+++ b/Objects/complexobject.c
@@ -78,7 +78,7 @@ c_quot(Py_complex a, Py_complex b)
      const double abs_breal = b.real < 0 ? -b.real : b.real;
      const double abs_bimag = b.imag < 0 ? -b.imag : b.imag;
 
-     if (abs_breal >= abs_bimag) {
+    if (abs_breal >= abs_bimag) {
         /* divide tops and bottom by b.real */
         if (abs_breal == 0.0) {
             errno = EDOM;
@@ -91,7 +91,7 @@ c_quot(Py_complex a, Py_complex b)
             r.imag = (a.imag - a.real * ratio) / denom;
         }
     }
-    else {
+    else if (abs_bimag >= abs_breal) {
         /* divide tops and bottom by b.imag */
         const double ratio = b.real / b.imag;
         const double denom = b.real * ratio + b.imag;
@@ -99,6 +99,10 @@ c_quot(Py_complex a, Py_complex b)
         r.real = (a.real * ratio + a.imag) / denom;
         r.imag = (a.imag * ratio - a.real) / denom;
     }
+    else {
+        /* At least one of b.real or b.imag is a NaN */
+        r.real = r.imag = Py_NAN;
+    }
     return r;
 }