diff --git a/diffrax/_step_size_controller/pid.py b/diffrax/_step_size_controller/pid.py
index 752b9bed..aa87bc39 100644
--- a/diffrax/_step_size_controller/pid.py
+++ b/diffrax/_step_size_controller/pid.py
@@ -487,7 +487,15 @@ def _scale(_y0, _y1_candidate, _y_error):
             _y1_candidate = jnp.where(_nan, _y0, _y1_candidate)
             _y = jnp.maximum(jnp.abs(_y0), jnp.abs(_y1_candidate))
             with jax.numpy_dtype_promotion("standard"):
-                return _y_error / (self.atol + _y * self.rtol)
+                _scale = self.atol + _y * self.rtol
+                if jnp.iscomplexobj(_y_error):
+                    # Divide the real and imaginary parts separately. A failed step
+                    # (e.g. an implicit solve that did not converge) reports
+                    # `y_error = inf`, i.e. `inf + 0j`. Complex division by `_scale`
+                    # would give `inf + nanj`, hence a NaN norm and a NaN step size,
+                    # and the solve would never recover.
+                    return lax.complex(_y_error.real / _scale, _y_error.imag / _scale)
+                return _y_error / _scale
 
         scaled_error = self.norm(jtu.tree_map(_scale, y0, y1_candidate, y_error))
         keep_step = scaled_error < 1
diff --git a/test/test_solver.py b/test/test_solver.py
index dfff3f6b..b7adad93 100644
--- a/test/test_solver.py
+++ b/test/test_solver.py
@@ -54,6 +54,49 @@ def test_implicit_euler_adaptive():
     assert out2.result == diffrax.RESULTS.successful
 
 
+@pytest.mark.parametrize(
+    "solver", (diffrax.ImplicitEuler(), diffrax.Kvaerno3(), diffrax.Kvaerno5())
+)
+@pytest.mark.parametrize(
+    "dtype, complex_dtype",
+    ((jnp.float64, jnp.complex128), (jnp.float32, jnp.complex64)),
+)
+def test_implicit_adaptive_complex_failed_step(solver, dtype, complex_dtype):
+    # `dt0=1` is too large for the root find, so the first step fails and reports
+    # `y_error=inf`. The step must be rejected and retried with a smaller `dt`. For
+    # complex `y` this used to give a NaN scaled error, hence a NaN `dt`, and the
+    # solve never finished. The complex solve should match the real one.
+    term = diffrax.ODETerm(lambda t, y, args: -10 * y**3)
+    t0 = jnp.array(0, dtype)
+    t1 = jnp.array(1, dtype)
+    dt0 = jnp.array(1, dtype)
+    tol = 1e-5 if dtype == jnp.float64 else 1e-3
+    stepsize_controller = diffrax.PIDController(rtol=tol, atol=tol)
+    sols = []
+    for y_dtype in (dtype, complex_dtype):
+        sol = diffrax.diffeqsolve(
+            term,
+            solver,
+            t0,
+            t1,
+            dt0,
+            jnp.array(1, y_dtype),
+            stepsize_controller=stepsize_controller,
+            max_steps=1000,
+            throw=False,
+        )
+        assert sol.result == diffrax.RESULTS.successful
+        assert sol.stats["num_rejected_steps"] > 0
+        sols.append(sol)
+    real_sol, complex_sol = sols
+    assert complex_sol.ys is not None and real_sol.ys is not None
+    assert complex_sol.stats["num_steps"] == real_sol.stats["num_steps"]
+    assert tree_allclose(complex_sol.ys.real, real_sol.ys)
+    assert tree_allclose(complex_sol.ys.imag, jnp.zeros_like(real_sol.ys))
+    true_y1 = jnp.array([1 / 21**0.5], dtype)
+    assert tree_allclose(real_sol.ys, true_y1, rtol=100 * tol, atol=100 * tol)
+
+
 class _DoubleDopri5(diffrax.AbstractRungeKutta):
     tableau: ClassVar[diffrax.MultiButcherTableau] = diffrax.MultiButcherTableau(
         diffrax.Dopri5.tableau, diffrax.Dopri5.tableau