As athletes after a substantial workout ice baths (54 degrees Fahrenheit and below) is good for the body to halt the excessive blood flow which you have induced through stress training. This allows for faster recovery and allows the body to recharge the nervous system for the subsequent day; in an extremely minimal fashion though.
Hot tubbing (or hot showers) increases blood flow and allows specifically the joints and small muscles to warm up for your upcoming level of stress which you put your body under. The circulation welcomes more fluid distribution in your joints and acts as a cushion for the stress (say in running, jumping, Olympic lifting, etc...). These are best done BEFORE you work out.
I have found from experience that the trick is being consistent in ice bathing after each workout and even on your off days to ice bath to speed recovery. The less inflamed a muscle is the greater the opportunity you have to work it out again. Hence people prescribe 48-72 hours of rest before the next same muscle group stimulation.
There is always the option of doing transition showers. These are hot and cold showers which shock the body. You can time your transition showers by doing 1 minute hot and 3 minutes cold/cool.
Honestly take a mildly warm shower after you workout. The extra sweat releases the toxins your body built up for the workout. See it as doing an extended cool down with stretching in a warm room. The muscles relax and the toxins of stress are released from the body. The cold shower won't make a substantial difference in your recovery unless you stand there for 20 minutes straight in freezing cold water.
As George Constanza put it so eloquently "Cold Showers? They're for psychotics".