This is a build on my comment, with the idea of using shishos (ladders) as computations. It is merely an attempt to give a computation model that is based on Go, and for which it makes sense to ask whether it is Turing-complete.
We start by fixing some powerful but not Turing-complete computation model that always halts, for instance some typed $\lambda$-calculus, or primitive recursive functions. This is the "cheating" part as we use an external computation model, but it is a weaker one, so hopefully it is the game of Go which will fill the gap and make it Turing-complete.
We consider an infinite goban labeled by $\mathbb Z\times\mathbb Z$.
Now the initial configuration of the goban can be infinite, and is given by an algorithm from our formalism: given coordinates $(i,j)$ the algorithm says whether the intersection $(i,j)$ is empty, occupied by a black stone, or occupied by a white stone.
We fix a maximal size $N$ of the groups in the initial configuration.
This means that given a position, we can always compute the group occupying it and its number of liberties, or answer "invalid configuration".
We also fix a marked stone (say black) at coordinate $(0,0)$.
An initial configuration is valid if at the beginning the group of the marked stone has two liberties.
Now we can view this configuration of the goban as the initial configuration of a non-deterministic machine, where a transition consists in playing a white stone at one of the two liberties of the marked group.
At each step, black automatically answers at the other liberty.
The run ends if
- The marked group is captured, in which case the input is accepted
- The marked group acquires more than $2$ liberties, in which case the input is rejected.
The run can also continue forever...
As for non-deterministic Turing machines, the input is accepted if there is an accepting run.
It is easy to simulate this machine with a non-deterministic TM, thanks to the existence of the bound $N$ (which is a parameter of the whole model).
Conjecture: For $N$ big enough (probably no more than $10$ to get the necessary gadgets), this model is Turing-complete.