Two different ways to calculate it without resorting to numbering like I did above:

Method 1

There are 361 intersections total.
1 of them (tengen) is unique
4x9 + 4x9 = 72 of them (the moves along midlines and diagonals) come in groups of 4
the other 361 - 1 - 72 = 288 come in groups of 8

So we have 1 + 72/4 + 288/8 = 55 unique moves.

Method 2

If we know that they come in a triangle shape, we can also calculate like this:
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = (1+10) + (2+9) + (3+8) + (4+7) + (5+6) = 5 x 11 = 55

Game tree complexity (of which initial number of moves is an input) and state space complexity are pretty standard, and go is indeed bigger than chess. But increased complexity doesn’t necessarily make a better game. Here’s a game with infinite complexity that’s really boring.

Player A thinks of any number.
Player B guesses what it was. If they are right they win this round, else they lose.

Swap roles and repeat an even number of times, tallying score.

(I brush over the difficulty of a human brain thinking of arbitrary transcendental numbers)

That other corner influence one corner for example…
Or the difficutie reading who has the better postion.
And laslty how mutch longer it needed until a computer defeaded a strong player