As Kevin Buzzard pointed out in the comments, the $j$-invariant has many possible normalizations.  Automorphisms of the affine line naturally act on the target space, so any $aj+b$ with invertible $a$ is a decent replacement.

That said, any normalization has infinitely many fixed points.  In fact, any choice of contractible fundamental domain for the action of $PSL_2(\mathbb{Z})$ has a fixed point in its closure.  You can prove this using Brouwer's fixed-point theorem together with the fact that the absolute value of $j$ increases exponentially toward cusps.

The $q$-expansion of $j$ converges reasonably quickly, so it is not hard to find fixed points numerically.  As far as I can tell, the fixed points don't have any particularly interesting arithmetic properties (but I would be interested to hear otherwise).