My first try: Let R^4=R+1 with R=-0.248126+1.03398*i and |R|=1.0633. Binary digits but forbid "11" and "101001" as substrings. (Are there any other forbiddances?) 1=0.0011(forbidden)=0.0001001001001...(allowed). Therefore it appears we can add and subtract. Tile T is tiled by T/R^2 (adjoin 10) and T/R^3 (adjoin 010). R*T is tiled by T/R and T/R^2. R^2*T is tiled by T and T/R. etc. Hence we apparently can tile the entire plane with just 3 tiles: T, T/R, and T/R^2.
--With this radix I think the allowed digit strings are got by repeatedly appending either 010 or 10 to the left of a digit string you already have. Then put decimal point anywhere. This whole quest is highly amenable to computer graphical and symbolical investigation. Neither of which I have done.