How to implement a custom layer wit multiple outputs in Keras?

In the call method of your layer, where you perform the layer calculations, you can return a list of tensors:

def call(self, inputTensor):

    #calculations with inputTensor and the weights you defined in "build"
    #inputTensor may be a single tensor or a list of tensors

    #output can also be a single tensor or a list of tensors
    return [output1,output2,output3]

Take care of the output shapes:

def compute_output_shape(self,inputShape):

    #calculate shapes from input shape    
    return [shape1,shape2,shape3]

The result of using the layer is a list of tensors. Naturally, some kinds of keras layers accept lists as inputs, others don't.
You have to manage the outputs properly using a functional API Model. You're probably going to have problems using a Sequential model while having multiple outputs.

I tested this code on my machine (Keras 2.0.8) and it works perfectly:

from keras.layers import *
from keras.models import *
import numpy as np

class Lay(Layer):
    def init(self):
        super(Lay,self).__init__()

    def build(self,inputShape):
        super(Lay,self).build(inputShape)

    def call(self,x):
        return [x[:,:1],x[:,-1:]]

    def compute_output_shape(self,inputShape):
        return [(None,1),(None,1)]


inp = Input((2,))
out = Lay()(inp)
print(type(out))

out = Concatenate()(out)
model = Model(inp,out)
model.summary()

data = np.array([[1,2],[3,4],[5,6]])
print(model.predict(data))

import keras
print(keras.__version__)