Custom gradient with complex exponential in tensorflow

As an exercise I am trying to build a custom operator in Tensorflow, and checking the gradient against Tensorflow's autodiff of the same forward operation composed of Tensorflow API operations. However, the gradient of my custom operator is incorrect. It seems like my complex analysis is not correct and needs some brushing up.

import tensorflow as tf

shape = (1, 16)
dtype = tf.complex64

x = tf.cast(tf.complex(tf.random.normal(shape), tf.random.normal(shape)), dtype=dtype)

def fun(x):
    phi = x * tf.math.conj(x)
    e = tf.exp(1j * phi)
    return e

def d_fun(x):
    d_phi = x + tf.math.conj(x)
    phi = x * tf.math.conj(x)
    d_e = 1j * d_phi * tf.exp(1j * phi)
    return d_e

@tf.custom_gradient
def tf_custom(x):    
    e = fun(x)
    def grad(dy):
        d_e = d_fun(x)
        return dy * d_e
    return e, grad

with tf.GradientTape() as g:
    g.watch(x)
    res = fun(x)
    
dy_dx = g.gradient(res, x)

with tf.GradientTape() as g:
    g.watch(x)
    res2 = tf_custom(x)
    
dy_dx2 = g.gradient(res2, x)

print(tf.reduce_sum(tf.abs(res - res2)).numpy())
print(tf.reduce_sum(tf.abs(dy_dx - dy_dx2)).numpy())