A full program for a WinForms-based app that draws a few colorful Drunkards' Walks.

ColoredWalker.cs

using System;
using System.Collections.Generic;
using System.Drawing;
using System.Windows.Forms;

namespace ColorWalker
{
    class Walker
    {
        private int _lastX = 0, _lastY = 0;
        private readonly int _maxVariance = 2, _speed = 3;
        private Color _current = Color.FromArgb(128, 128, 128);
        private Pen _artMaker = new Pen(Color.White, 3);
        private Random _rng;

        public Walker(Random rng, int startingX, int startingY)
        {
            this._rng = rng;
            this._lastX = startingX;
            this._lastY = startingY;
        }

        private bool Wrap(ref int what, int min, int max)
        {
            if (what < min)
                what = max;
            else if (what > max)
                what = min;
            else
                return false;
            return true;
        }

        public Tuple<int, int, Color> UpdateAndDraw(Graphics g, Size clientSize)
        {
            int nextX = _lastX, nextY = _lastY;
            nextX += _rng.Next(-_speed, _speed + 1);
            nextY += _rng.Next(-_speed, _speed + 1);
            bool centered =
                Wrap(ref nextX, 0, clientSize.Width) | // no, I didn't mean ||
                Wrap(ref nextY, 0, clientSize.Height);
            int nextR = _current.R + _rng.Next(-_maxVariance, _maxVariance + 1);
            Wrap(ref nextR, 0, 255);
            int nextG = _current.G + _rng.Next(-_maxVariance, _maxVariance + 1);
            Wrap(ref nextG, 0, 255);
            int nextB = _current.B + _rng.Next(-_maxVariance, _maxVariance + 1);
            Wrap(ref nextB, 0, 255);
            _current = Color.FromArgb(nextR, nextG, nextB);
            _artMaker.Color = _current;
            if (centered)
            {
                g.DrawEllipse(_artMaker, nextX, nextY, 1, 1);
            }
            else
            {
                g.DrawLine(_artMaker, _lastX, _lastY, nextX, nextY);
            }
            _lastX = nextX;
            _lastY = nextY;
            return Tuple.Create(nextX, nextY, _current);
        }
    }

    public partial class ArtMakingForm : Form
    {
        #region Windows Form Designer generated code
        /// <summary>
        /// Required designer variable.
        /// </summary>
        private System.ComponentModel.IContainer components = null;

        /// <summary>
        /// Clean up any resources being used.
        /// </summary>
        /// <param name="disposing">true if managed resources should be disposed; otherwise, false.</param>
        protected override void Dispose(bool disposing)
        {
            if (disposing && (components != null))
            {
                components.Dispose();
            }
            base.Dispose(disposing);
        }

        /// <summary>
        /// Required method for Designer support - do not modify
        /// the contents of this method with the code editor.
        /// </summary>
        private void InitializeComponent()
        {
            this.components = new System.ComponentModel.Container();
            this.timer = new System.Windows.Forms.Timer(this.components);
            this.SuspendLayout();
            // 
            // timer
            // 
            this.timer.Enabled = true;
            this.timer.Interval = 1;
            this.timer.Tick += new System.EventHandler(this.Stumble);
            // 
            // ArtMakingForm
            // 
            this.AutoScaleDimensions = new System.Drawing.SizeF(6F, 13F);
            this.AutoScaleMode = System.Windows.Forms.AutoScaleMode.Font;
            this.ClientSize = new System.Drawing.Size(496, 473);
            this.Name = "ArtMakingForm";
            this.Text = "Watch The Colorful Drunk";
            this.Load += new System.EventHandler(this.ArtMakingForm_Load);
            this.Paint += new System.Windows.Forms.PaintEventHandler(this.Stumble);
            this.ResumeLayout(false);

        }

        private System.Windows.Forms.Timer timer;
        #endregion

        private Random _rng = new Random();
        private List<Walker> _walkers = new List<Walker>(10);
        private Dictionary<Tuple<int, int>, Color> _drawnBefore =
            new Dictionary<Tuple<int, int>, Color>();

        public ArtMakingForm()
        {
            InitializeComponent();
            this.ClientSize = new Size(512, 512);
            this.MaximumSize = this.Size;
            this.MinimumSize = this.Size;
            for (int i = 0; i < _walkers.Capacity; ++i)
            {
                _walkers.Add(new Walker(_rng, this.ClientSize.Width / 2,
                    this.ClientSize.Height / 2));
            }
        }

        private void ArtMakingForm_Load(object sender, EventArgs e)
        {
            timer.Start();
        }

        private void Stumble(object sender, EventArgs e)
        {
            Graphics g = this.CreateGraphics();
            foreach (var walker in _walkers)
            {
                var res = walker.UpdateAndDraw(g, this.ClientSize);
            }
        }

        [STAThread]
        static void Main()
        {
            Application.EnableVisualStyles();
            Application.SetCompatibleTextRenderingDefault(false);
            Application.Run(new ArtMakingForm());
        }
    }
}