While multi-brain and multi-hemisphere answers have been posed, they all result in some loose of function while one part of the cognitive system is down. This seems against what the OP is looking for since he does not want dolphin style torpor.
However, computer scientists have already devised a number of systems called RAID controllers to be used by servers to replicate data across multiple hard drives in case of a disk failure and to allow for scheduled maintenance without downtime. When you consider sleep as analogous to server maintenance, it becomes obvious that a brain could use a similar system to overcome the need for sleep with a somewhat larger brain that functions as 2 identical brains. By replicating functionality, when one brain is ready to sleep, it wakes up the other brain and duplicates it's memories of the day to it, and then goes to sleep. In this fashion, you would not notice a performance drop-off or change in personality because both brains are doing the same things with the same hardware and the same information, just at different times of day.
So what does this look like on a brain? During the day you are bombarded by more more information than could be stored in your entire brain, but we still have long term memories that go back for years because our hippocampus processes all of this information and reduces it to a few trivial bits of information that it writes to long-term memory. Now imagine two brains connected by a new organ at the hippocampus. Let's call this new organ your raid ganglion. Your raid ganglion will work like a cache, a small and efficient place designed to store the memories you've acquired in a day. Each time your hippocampus writes a long term memory to the active brain, it also writes the instructions for creating that memory to the inactive brain in the raid ganglion. The raid ganglion retains these instructions until the sleeping brain wakes up (so as not to disturb the sleeping operations), and then dumps all the memories all at once into the second brain. The the raid ganglion then goes through its own sort of sleep cycle to purge its memories while both brains are active recording things in stereo. Once the raid ganglion is purged, the first brain can go to sleep and the second brain can begin using it to store its memories of the day.
As various comments and other answers have pointed out, there are several possible concerns here that require elaboration:
1. Sleep is poorly understood
The nature of sleep is generally irrelevant for this answer. We do not need to understand much about how sleep works to understand that it is an "offline process" where the brain is doing something other than what it does while you are awake. Whether it is a time to sort memories, replenish neurotransmitters, repair damage, etc. is inconsequential. As long as it is an isolated enough system to do what the brain is designed to do while sleeping, it can do that while the second brain takes over.
2. Consciousness is poorly understood
This is a more existential question than anything. To the outside observer, two brains attached to the same body with all the same capabilities, memories, skills, endocrine system, etc. would be indistinguishably the same person. If you ask either brain the same question, they will generally come to the same answer as long as they are syncing up their memories every switch over, and neither of them is significantly damaged.
From the inside, you would technically only "be you" half of the time, but you would always receive all of the memories and experiences as the other you. When "you" are in control, you really would not be able to tell the difference between when were awake, and when you just remember being awake. You might experience something like deja vu twice a day for those moments when both minds are trading off, but the human mind is full of filters designed to merge multiple inputs into a single stream of consciousness; so, a species designed to function this way would likely have a mechanism for making the switch over feel seamless the same way we can have 1 eye open, then open our 2nd eye, then close our first eye and feel like we are experiencing a continuous view of what we are looking at.
3. Removing sleep makes our bodies less optimised
This becomes a matter of perspective that we can again reference how RAID servers work to gain more insight. RAID servers always need more hardware than an equally powerful desktop computer. It needs more disk space, more wiring, more data processing, etc. A RAID server can cost several times as much as an equivalently powerful PC, so why do we use them at all? The answer is because they are optimized for something other than cost. A RAID server can maintain much better uptime and information preservation than a traditional computer. If part of a RAID brain gets damaged, it can potentially heal the damaged part of the brain using the duplicate structure to fully restore the lost memories. If the organism needs to think around the clock more than it needs to be able to be motile, then the bigger, heavier, brain would be more fit. We as humans use the same optimisations that were relevant to our ancestors, but after a few million years of civilization, or a few rounds of genetic engineering, humans could find RAID brains to be optimal in our environment where our greatest threats are the ravages of age and mental competition.
4. RAID brains are just a less optimal version of having two independant brains
It is true that you will need more neural structures operate a RAID brain than two independent brains, but this again comes down to what you are optimising for. The point here is not just no sleep, but no sleep without loss of function. The problem with two brains and one body is that both brains will have different experiences based when they have been awake. So, they develop different fears, values, skills, etc. If one decides to flee and one decides to fight, then neither strategy will work and your creature will die. If a RAID brained animal is threatened, during a time both brains are active, they will both generally react in unison instead of against each other because they have the same experiences.
As for what happens when only one brain is awake, both animals will be able to act in a single minded fashion, but the brain that does not share experiences will be unable to use what it's learned during one time of day to function at the other. This means the animal will often have to learn the same thing twice. Not only does this waste a lot of time much like sleeping does, but it exposes the organism to twice as much danger. If a one-brained or RAID brained animal does something stupid that almost kills it, then it learns from its near death experience and never does that again. If a 2 brained animal does something like that at night, then during the day it does the same thing, it has twice as many chances to die.
So, a 2 brained animal is in many ways less optimal than a 1 brained or RAID brained animal.