The following code snippet lead to some surprises:

import numpy as np
from sklearn.datasets import load_iris
from sklearn.impute import SimpleImputer

X, y = load_iris(return_X_y=True)
X[:, 0] = np.nan

imputer = SimpleImputer(keep_empty_features=False, strategy="constant", fill_value=1)
X_trans = imputer.fit_transform(X)

assert X_trans.shape[1] == 3, f"X_trans contains {X.shape[1]} columns"

---------------------------------------------------------------------------
AssertionError                            Traceback (most recent call last)
Cell In[19], line 11
      8 imputer = SimpleImputer(keep_empty_features=False, strategy="constant", fill_value=1)
      9 X_trans = imputer.fit_transform(X)
---> 11 assert X_trans.shape[1] == 3, f"X_trans contains {X.shape[1]} columns"

AssertionError: X_trans contains 4 columns

Apparently this is something that we really wanted for backward compatibility when merging #24770:

Now, I'm wondering if we should not deprecate this behaviour since the parameter keep_empty_feature allows to control whether or not we should drop the feature entirely.

So I would propose to warn for a change of behaviour when strategy="constant", keep_empty_feature=False, and that we detect that we have empty feature(s).