C (gcc), 139 bytes

Compile with -Dk="_nZZiaeY" (included in byte count). -2 bytes if prompt >>\n is allowed.

x;f(c){for(printf(">>");c=getchar()-10;x+=c--?c--?c--?c||printf("%i\n",x),0:x*x-x:-1:1,x*=~x&&x^256)c=strchr(k,c)-k>>1;f();}

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Degolf

/** Preprocessor **/
-Dk="_nZZiaeY" // This is a lookup string; it corresponds to "ixddskoc",
               // with 10 deducted from each character. Upon compilation, 
               // all occurences of the string literal are replaced with a 
               // pointer to its location in memory.

/** Source **/
x;f(c){ // x: 32-bit accumulator, c: local variable for read character
    for(printf(">>"); // Start for-loop and print prompt.
            c=getchar()-10; // Read a character from STDIN.
                            // Loop breaks if it is '\n'.
            
            // The below happens at the end of each iteration.
            x+=c--?c--?c--? 
               // Find the correct operation by testing c and post-
               // decrementing with multiple ternary-ifs. If c is 0, 
               // operation was found, add the else-value to the 
               // accumulator x.
               //     If the character is invalid, the value of c is
               // very large, and will not reach 0 with 3 decrements.
                  
               c||printf("%i\n",x),0 
               // If c-3 is 0, print accumulator, else do nothing.
               // Returns 0 regardless of what happens. (No change to x)
               :x*x-x 
               // Square. Results in x=x+x*x-x, and is shorter than (x*=x)
               :-1:1, 
               // Decrement, Increment.
               x*=~x&&x^256 
               // Because -1==0xffffffff, ~x==0 when x==-1. Likewise,
               // x^256==0 only when x==256. The logical-AND coerces the result
               // to boolean 1 (no match) or 0 (match). Multiplication resets
               // the accumulator as appropriate.
           )
        // This is the actual body of the for-loop
        c=strchr(k,c)-k>>1; 
           // Finds the index of the read character in the lookup string,
           // then "divides" it by two.
           // Because strchr() returns NULL (0) when character is not found,
           // deducting k from it results in a very negative number.
           // The right-shift results in division by 2 for positive numbers, 
           // while the negative numbers become very large positive numbers
           // (c >= 0x70000000) because of the 2's complement representation.
    // Finally, recurse until forceful termination.
    f();
}