Efficient way to implement an indexed queue (where elements can be retrieved by index in O(1) time)?

You can use a cyclic array. I.e. implement queue in array.

The implementation is pretty trivial, you don't need to use external library, just implement it yourself. A hint: it's easier to use m_beginIndex, m_nElements members than m_beginIndex, m_endIndex.

public class IndexedQueue<T>
{
    T[] array;
    int start;
    int len;

    public IndexedQueue(int initialBufferSize)
    {
        array = new T[initialBufferSize];
        start = 0;
        len = 0;
    }

    public void Enqueue(T t)
    {
        if (len == array.Length)
        {
            //increase the size of the cicularBuffer, and copy everything
            T[] bigger = new T[array.Length * 2];
            for (int i = 0; i < len; i++)
            {
                bigger[i] = array[(start + i) % len];
            }
            start = 0;
            array = bigger;
        }            
        array[(start + len) % array.Length] = t;
        ++len;
    }

    public T Dequeue()
    {
        var result = array[start];
        start = (start + 1) % array.Length;
        --len;
        return result;
    }

    public int Count { get { return len; } }

    public T this[int index]
    {
        get 
        { 
            return array[(start + index) % array.Length]; 
        }
    }        
}