Blurple Malzahar.
First of all... blurple? REALLY DUDE?! Yeah I know! I'm bad at coming up with names, sry!
What is blurple Malzahar? Blurple malzahar consists of a semi-hybrid, ad focused Malzahar build which makes you an an-ti tank duelist assasin mage fighter splitpusher. No, I'm not lying, it does all of these things. It is literally a build I came up with in an ARAM match which I tested in custom games and am now going to test in PVP games
Let's get into the concepts of this build:

These are your 4 core items. It makes you a monster hybridly while giving some nice defensive stats. I would throw in a RoA but I'm not 100% sure it will fullfill the purpose of this build, right now I'm testing it with these 4 items only, then going to see which items go nicely. The point of this is to give you the highest amount of pushing power, dueling power and still make you relevant.

Manamune, your first core item for Blurple Malzahar and you should be rushing this. The reason why this item will fit so well is mainly because of the AD it gives you and your creeps, the mana and mana regen + the active that will help you a lot late game. This solves most of your mana problems, so you shouldn't worry about another Mana item for a while.

Your source of Damage come mid game. This gives you more AD, AP for the nice scalings, Lifesteal, Spell vamp and an active that slows and deals damage. This is your bread and butter for offensive stats + sustain, it is so important to get this item second because if you go another item, your AD or AP will be too weak or too heavy, you will have very low sustain through 1 source AND you will miss out on the active.
Whether you get
or
first depends on how you are doing, what you deal the most damage with at the moment and how you plan on sustaining. I would prefer to go
first, but sometimes the active from
is just what you need.

3rd/4th item. You should get this if you have an AP match-up and they can silence/disable/cc you, as it will give you tons of AD, nice magic resist and a shield that can save your life. It is really important to realize that your creeps deal half of your AD damage, and because you can spam abillities by now, you will have them up more often than not. It's also what makes this build hybrid, because otherwise you would opt for
instead of
.

3rd/4th item. Again, get this if you have an assassin/ad matchup. It is what will save your life when you get caught/engaged on in a teamfight/duel, what will give you your other half of magic damage. I choose this over
because you need the defensive stats/active due to having unreliable escapes. Flash is more than enough most of the time, but against cc and mobility you cannot simply keep up.
Other items to be considered:
It applies it's passive with your creeps, it gives you ad, cdr, health and armor shred. It will be a good item against anything, the total damage output is insane, but I did not put it in the list because it does not build well with Malzahar as far as the lesser items go, and costs a lot of gold for the stats it could be giving him otherwise. Situational, 2nd or 3rd item.
I would either take this as my 3rd item or my 5th item, as your 4th item you want to be your AD item. I have faith this item will go as 5th item in this build, maybe 3rd/4th replacing Zhonya, but I'm still debating it.
This is either my 4th/5th/6th item. If I take it 4th I would replace it with Malmortius. The reason this item MIGHT work is because of lack of cleanse/Zed/hard cc. It gives you nice AD (= to Infinity Edge) and magic resist, + a free cleanse.
I would choose this if you are more of a supportive role trying to deny tanks/splitpush as it gives your team MR, powers up a minion(not sure if it works with your own creep) and gives AP.
An item considered core on Malzahar, not sure if great choice due to slot syndrome, but can work I suppose.
Probably an option to replace
if you need the ad/attackspeed and the active. I would pick this if your team lacks ad damage(top is tank/ap, jungler is tank/ap and bottom is tank/ap). Might end up replacing it overall.
/
If their team stacks a lot of 1 resistance, then this might be a good 3rd/4th/5th.
Your abillities:
Q: A ranged silence. That's all there is to it. Deals nice damage, use this to stop casters/cc lockers from shredding your team. Max this 2nd/3rd to increase it to a 3 second silence.
W: AoE pool of despair that does nice damage against anything. It deals around 10-12% of target's max health as magic damage come mid-late game. People underestimate this ability so much, it's ridiculous. Max this last, as the scaling per rank is not as nice.
E: Contrary to popular belief: This is not an ignite. This is your bread and butter for damage, harass, duel, anything. It makes your creeps go HAM on the target, does 320 base damage at max rank, and if it kills a unit, gives you mana AND spreads to nearby units. You want to use this to push the wave consistently against champions that have a hard time farming. Yes, it can be used to execute enemies. Remember if you use it wisely you could have up to 2 of these going at the same time. The horrors... Max this 2nd/3rd.
R: A ranged
ult that deals 550 base damage at rank 3? Pls. Just be careful you don't accidentally cancel it by moving or attacking, as it can matter, A LOT. The damage is dealt over time every 0.5 secs, not instantly - same as your E. Max this 1st.
Purpose of Blurple:
Your role is to be an asset to your team while dealing hybrid damage. You will focus a little on the AD side most of the time because your scaling with AP and base damages are insane, so because of this you do not put a lot into AP,since you already deal decent damage. Does your team need a splitpusher? Grab a banner, some ad and you got yourself a pusher. Need a strong duelist/anti tank? Your mix of damage already does that for you. Your W shreds tanks, E is OP and your ult shuts down assassins. Need a ranged poker? Get a luden's echo, and laugh as they eat your Qs.
You shoud Max:
R>E>Q>W usually. R>Q>E>W agaisnt casters/mages.
Summoner Spells:
is a must, although you could try swapping it for
and
are good counters to assassins/mages/carries/cc and give you survivability. Take this if you want to play passive/fear dying.
if you feel offensive, and have a decent matchup.
if you are going toplane/need to be around the map.
if you are jungling, though again, not designed for the jungle. Try at your own risk. If you do, I suggest taking
or 
Downsides of this build: You don't deal 1 type of damage, therefore your damage output is not focused on 1 type, being generally bad. It is still a high output though. Without health items to back you up (except RoA), this build has the downfall that if you get bursted/focused, you will go down quickly. Resistances do help, but health is equally important. You can confuse your team/make them think you are a troll. It is not tested - use it at your own risk. You are still immobile for the most part.
Blurple is just pure theorycraft for the most part, as again, I have not tested in PVP. Feel free to give feedback. AND PLEASE DON'T TRY THIS IN RANKED.
Thank you.