I feel like you're mostly thinking about the conscious mind. However, the mind has a much stronger subconscious part. This is well evident in how it processes things. Every day your subconscious is filtering through your daily life, ordering all visual, tactile, proprioceptive, taste, hearing and smelling stimuli. This creates both a conscious and subconscious view of the world. It pushes some things to the conscious, like your name over the din of a crowd, or a familiar face, or a hamburger joint while you’re very hungry. Subconsciously you’re still processing a lot more, from idea’s like what to write to that mean poster on the internet three days ago, to what car you should buy, or simply driving you home in the car while your attention was completely somewhere else.
Consciousness can in a very crude way be seen as a filter on the subconscious. Anything deemed important enough is pushed to the consciousness, which will allow it to be more easily stored in memory and reacted to more actively.
The question then is, why can’t we already multitask incredibly well? First of all, as the most intelligent species of the planet, we can. Just not as well as you would like as an answer. The part of the brain that gives us our intelligence is relatively young evolutionary speaking, which is very visible if compared to the limbic system, cerebellum and brain stem. The density of these older systems of the brain is much higher and has more surface area thanks to sulci and gyri (what makes the brain look like a wrung out sponge). Our current intelligence is simply fledgeling and only just capable of sustaining the amazing things we can do currently. It can even be seen in the ‘leaking’ of information from one part of the brain to the other, where stimuli of for example a body part can leak into the neighbouring brain area, causing you to feel some things there as well.
How would we improve? Simply by making it more densely packed and larger, with more sulci and gyri. This way, we’ll be able to process much more information with less leaking, as long as the nutritional supply is enough and further strain to each brain volume isn’t increased. Otherwise many stress symptoms can be seen quite quickly, or simply shut down or death from malnutrition. Increasing your brain volume will also increase brain power, allowing to process more, deeper and faster on both a conscious and unconscious level.
However, we have a second barrier we already deal with every day. Not only is the brain holding us back, our physical body does so as well. Our eyes have only a small focal area, meaning you simply can’t see four screens effectively at once. Even rapid switching from screen to screen will make you drop a lot of information in the saccade (quick movements) of the eyes. Our smell and hearing are very limited compared to a lot of other species. Our hands do have an insane amount of tactile sensors, but most of our body doesn’t. Many parts even share sensory input, like the back that has many tactile sensors that are registered on the same neuron to the brain. This obfuscates many information, which is fine for our lives but maybe not so much for your super humans. If the brain capacity has been increased, more stimuli can be processed as well.
In short, our brain is already impressive, yet fledgling. Making it larger and more developed in density and surface area will help a great deal. In addition, improve the body to increase the capable stimuli density. Otherwise you’ll never be able to have the required input for your desired multi tasking.
As one addendum, I think no ‘second’ conscious is possible. However it’s damn hard to imagine with only one consciousness, so how can we know? For all we know our subconscious is much like our normal consciousness, just It’s very good at keeping temperature, heartbeat, breathing and the odd movement of an arm or so.