I expanded it out and got $e^{4z+1} = e^{4x+1}\cos{4y} + e^{4x+1}\sin{4y}$
Then my CR equations were -
$U_x = (4x+1)e^{4x+1}(4)(\cos 4y)$
$U_y = -e^{4x+1}(\sin4y)$
$V_x = (4x+1)e^{4x+1}(4)(\sin 4y)$
$V_y = e^{4x+1}(\cos 4y)$
Taking $U_x = V_y$ I get
$(4x+1)e^{4x+1}(4)(\cos 4y) = e^{4x+1}(\cos 4y)$
$(4)(4x+1) = 1$
But that can't be right as then it means the CR equations are only satisfied for a certain value of x. So what am I doing wrong?