First I want to know.. where is it mentioned that this is true for dimension 1?
[disregard this] To spoil the fun, i suggest us to look at the case when g is the identity (if I understood by "commuting" you meant that the composition of f and g commutes).. I find to find a counterexample.. so I reduce the question to... is there a continuous function f from unit ball to unit ball such that f(x) ≠ x for all x in the unit ball.. Intuition tells me that such a function indeed exists. Consider n=1 and now look at the graph of g(x) = x and consider the box [-0.5,0.5] x [-0.5,0.5], you can easily "draw" a function that lies below the diagonal {(x,x) : x in [-0.5,0.5]} that does not touch the diagonal. So I'm not sure if the case n=1 even holds. Should there be an extra condition here or am I missing something? [disregard this]
I made a little google search and found this paper, that mentions that the conjecture holds true for polynomials and special functions Cohen calls "full functions". I'm not able to download the paper though. Cohen's paper is: H. Cohen, On fixed points of commuting functions, Proc. Amer. Math. Soc. 15. (1964),