  An involution of a real commutative algebra $A$ is a real-linear homomorphism
$f : A \rightarrow A$ such that $f^2 = \mathrm{Id}$. We show that there are six
involutions of the algebra of bicomplex numbers, contrary to the actual number
of four stated in the literature. We also characterize $n$-involutions
satisfying the additional property $f^n = \mathrm{Id}$ for some integer $n \geq
2$. We show there are eight $n$-involutions and they occur only for $n = 2$ and
$n= 4$. We use our result to give a new characterization of the invertible
elements of the algebra of bicomplex numbers.
