You could move some of the cost to index-creation time by converting the per-row indices into flattened indices:

In [1]: a = np.random.random((5, 6))                                            

In [2]: i = a.argmax(axis=1)                                                    

In [3]: a[np.arange(len(a)), i]                                                
Out[3]: array([0.95774465, 0.90940106, 0.98025448, 0.97836906, 0.80483784])

In [4]: f = np.ravel_multi_index((np.arange(len(a)), i), a.shape)              

In [5]: a.flat[f]                                                              
Out[5]: array([0.95774465, 0.90940106, 0.98025448, 0.97836906, 0.80483784])


I haven't benchmarked, but I suspect this will be faster if you're using the same index multiple times.