diff --git a/pandas/plotting/_matplotlib/compat.py b/pandas/plotting/_matplotlib/compat.py
index 964596d9b6319..729d2bf1f019a 100644
--- a/pandas/plotting/_matplotlib/compat.py
+++ b/pandas/plotting/_matplotlib/compat.py
@@ -22,3 +22,4 @@ def inner():
 mpl_ge_3_1_0 = _mpl_version("3.1.0", operator.ge)
 mpl_ge_3_2_0 = _mpl_version("3.2.0", operator.ge)
 mpl_ge_3_3_0 = _mpl_version("3.3.0", operator.ge)
+mpl_ge_3_4_0 = _mpl_version("3.4.0", operator.ge)
diff --git a/pandas/plotting/_matplotlib/tools.py b/pandas/plotting/_matplotlib/tools.py
index 955a057000c41..da7b451137c94 100644
--- a/pandas/plotting/_matplotlib/tools.py
+++ b/pandas/plotting/_matplotlib/tools.py
@@ -366,6 +366,11 @@ def handle_shared_axes(
             row_num = lambda x: x.rowNum
             col_num = lambda x: x.colNum
 
+        if compat.mpl_ge_3_4_0():
+            is_first_col = lambda x: x.get_subplotspec().is_first_col()
+        else:
+            is_first_col = lambda x: x.is_first_col()
+
         if nrows > 1:
             try:
                 # first find out the ax layout,
@@ -397,7 +402,7 @@ def handle_shared_axes(
                 # only the first column should get y labels -> set all other to
                 # off as we only have labels in the first column and we always
                 # have a subplot there, we can skip the layout test
-                if ax.is_first_col():
+                if is_first_col(ax):
                     continue
                 if sharey or _has_externally_shared_axis(ax, "y"):
                     _remove_labels_from_axis(ax.yaxis)
diff --git a/pandas/tests/plotting/frame/test_frame.py b/pandas/tests/plotting/frame/test_frame.py
index 68e693cdb85e2..baed44e09b581 100644
--- a/pandas/tests/plotting/frame/test_frame.py
+++ b/pandas/tests/plotting/frame/test_frame.py
@@ -697,7 +697,9 @@ def test_plot_scatter_with_categorical_data(self, x, y):
 
         _check_plot_works(df.plot.scatter, x=x, y=y)
 
-    def test_plot_scatter_with_c(self):
+    def test_plot_scatter_with_c(self, request):
+        from pandas.plotting._matplotlib.compat import mpl_ge_3_4_0
+
         df = DataFrame(
             np.random.randn(6, 4),
             index=list(string.ascii_letters[:6]),
@@ -709,9 +711,10 @@ def test_plot_scatter_with_c(self):
             # default to Greys
             assert ax.collections[0].cmap.name == "Greys"
 
-            # n.b. there appears to be no public method
-            # to get the colorbar label
-            assert ax.collections[0].colorbar._label == "z"
+            if mpl_ge_3_4_0():
+                assert ax.collections[0].colorbar.ax.get_ylabel() == "z"
+            else:
+                assert ax.collections[0].colorbar._label == "z"
 
         cm = "cubehelix"
         ax = df.plot.scatter(x="x", y="y", c="z", colormap=cm)