I have a numpy array of arbitrary shape, e.g.:
a = array([[[ 1, 2],
[ 3, 4],
[ 8, 6]],
[[ 7, 8],
[ 9, 8],
[ 3, 12]]])
a.shape = (2, 3, 2)
np.argmax(a, axis=1) = array([[1, 1, 0],
[1, 0, 1]])
np.max(a, axis=1) = array([[ 2, 4, 8],
[ 8, 9, 12]])
a[np.arange(len(a)), np.argmax(a, axis=1)]
IndexError: shape mismatch: indexing arrays could not be broadcast together with shapes (2,) (2,3)
You can use advanced indexing

In [17]: a
Out[17]:
array([[[ 1, 2],
[ 3, 4],
[ 8, 6]],
[[ 7, 8],
[ 9, 8],
[ 3, 12]]])
In [18]: idx = a.argmax(axis=1)
In [19]: m,n = a.shape[:2]
In [20]: a[np.arange(m)[:,None],np.arange(n),idx]
Out[20]:
array([[ 2, 4, 8],
[ 8, 9, 12]])
For a generic ndarray case of any number of dimensions, as stated in the comments by @hpaulj
, we could use np.ix_
, like so 
shp = np.array(a.shape)
dim_idx = list(np.ix_(*[np.arange(i) for i in shp[:1]]))
dim_idx.append(idx)
out = a[dim_idx]