So I was curious what would be the advantages and disadvantages to have 12 DNA bases? For example could it be used to store complex memories in an organism, could it cause very complex structure in an organism that wouldn't be possible in ones with only 4, would it cause more cancer or cancer like diseases, etc...?
It wouldn't provide all that much of an advantage.
DNA basically stores a string of data. We can measure that in bitrates. Our current system with 4 bases can store 2 bits per nucleotide. For example, we can say A=00, C=01, G=10, and T=11.
A 12 based DNA is a bit trickier to count. If we skip to a 16 symbol system first (because it's easier) we can see that that would store 4 bits per nucleotide: A=0000 B=0001 C=0010 D=0011 E=0100 F=0101 G=0110 H=0111 I=1000 J=1001 K=1010 L=1011 M=1100 N=1101 O=1110 P=1111.
The actual relationship is the bits-per-symbol is $\log_2S$ where $S$ is the number of symbols you have (i.e. number of individual nucleotide). We can use that to see that this 12-base DNA would be able to encode 3.58 bits/nucleotide, or just shy of twice as much data we can store.
This means their genetic material can encode data in half as many symbols as we need. Not all that impressive. Worse, it's not guaranteed that that means you actually take up half as much space. DNA is pretty simple. A 12 base system might have larger nucleotide, and take up more space!
Replication errors would also be more difficult to deal with, because there's simply more possibilities for what the data should have looked like.
The one really interesting thing you could explore with this 12-base system is how the DNA interacts with things. Consider that we have 22 amino acids that we build our proteins from. We only need 4 nucleotide, but 22 amino acids. Why? Because the amino acids have to do something. They have chemical characteristics that matter when we make proteins, beyond just their ability to hold onto data.
You might be able to explore a creature that finds it valuable to have the DNA directly interact with the environment, and thus more bases means more ways it can interact.
One of the many hypotheses about life forming on Earth is that a single molecule acted as both a data storage device (like DNA/RNA) and an actor on the chemical scene (like a protein). Your 12 base system might be the result of that system taking a very different track in the first half billion years of life.
While from an information theoretical point of view the number of bases is irrelevant (as long as there are at least two of them), a larger number of bases allows for a more dense packing of the genetical information, using shorter strangs of DNA, consuming less (potentially rare) phosphorus, or allowing for more luxurious proteins with more than the usual 20 plus 2–4 aminic acids.
On the downside, there is a more complex chemistry to be handled, more chances of transcription errors, and a higher rate of mutations. Finding 6 base pairs with nice matchings may also turn out difficult (synthetic biologist are experimenting with some unnatural base pairs but they are far away from having six of them.