Mahadev makes Goddess Lakshmi realise that she has done a mistake by
suspecting Lord Vishnu. Goddess Lakshmi pleads with Mahadev to bring
Lord Vishnu back. Mahadev goes to the Patal Lok in Vrishabh’s disguise
to bring Lord Vishnu back. Mahadev and Lord Vishnu fight with each
Their fight continues for years. Ganesha promises Parvati to
put an end to Mahadev and Lord Vishnu’s fight. Lord Brahma becomes
worried as Lord Vishnu attacks Mahadev with Narayana Astra.
There are multiple incidents you could be referring to:.
The story depicted in the TV serial you linked to is (loosely) based on a story told in this excerpt from the Shatarudra Samhita of the Shiva Purana, about a battle between Vishnu and Shiva's incarnation Vrishabha the bull, AKA Vrishesha.
Most accounts of Churning of the Ocean end with Vishnu tricking the Asuras by turning into Mohini, and then the gods defeating the Asuras (demons) in battle after drinking the Amrita (nectar of immortality). But the Shiva Purana tells a modified version of the story, where a bunch of beautiful Apsaras (nymphs) are born out of stray drops of Amrita, and the Asuras enchanted by their beauty kidnap them and take them to Patalaloka (the underworld). Then the battle between the gods and the Asuras take place, and Vishnu in the course of fighting the Asuras chases them all the way to Patalaloka. But then after he defeats them, he sees the beautiful Apsaras, and struck by the arrow of Kama (the god of love) he becomes enamoured of them. So he fathers a bunch of children with them, and those sons soon becomes a menace to the three worlds:
Vishnu begot of them sons of great exploits and valour, experts in various kinds of warfare, shaking the entire earth. Those sons of Vishnu of great strength and valour wrought great havoc both in heaven and earth causing misery to all. On seeing the geat harm done to the worlds the gods and the sages approached Brahma and informed him of this after bowing to him.
Then Brahma goes to Shiva, informing of him of the mayhem caused by the sons of Vishnu, and asks Shiva to make Vishnu leave Patalaloka and stop his dalliances with those Apsaras. So Shiva turns into Vrishabha the bull and goes into Patalaloka, killing Vishnu's sons and fighting Vishnu:
Then bellowing in the form of a Bull, [Shiva] the Pinaka-bearing lord entered the crevice and shouted terribly. At his bellowing sound, cities and towns crumbled down and all the citizens trembled. Then the bull attacked Vishnu's sons wielding their bows for fighting, deluded by Siva's Maya and proud of their great strength and exploit.... [T]he sons of Vishnu became furious. Those heroes roared loudly and rushed against Siva. Rudra in that form of a bull was furious at those sons of Vishnu who rushed at him. He kicked them with his hoofs and tore them with his horns. When their limbs were split by Rudra, they swooned and lost their lives. They were immediately destroyed.
When they were killed Vishnu the foremost of the strong came out shouting loudly and hastened towards Siva. On seeing Siva in the form of a bull going along after killing his sons, Vishnu hit him with divine arrows and
missiles. Then the infuriated Siva, the highly powerful one in the form of a bull stood like a mountain and caught hold of all those missiles of Vishnu. O sage, shouting furiously, lord Siva in the form of a bull shook the three worlds terribly. The lord leaped up suddenly and kicked the infuriated Vishnu who was deluded and could not realise him. The lard then tore him with his horns. Unable to bear the onslaught, Vishnu deluded by Maya was exhausted and utterly shattered.
So Vishnu finally snaps out of the Maya and realizes that the bull is Shiva. Embarrassed that he didn't recognize Shiva sooner, Vishnu leaves his Sudarshana Chakra (discus weapon) in Patalaloka and returns to Lakshmi, and Shiva provides him with a new Sudarshana Chakra, even more powerful than the old one. (The Sudarshana Chakra was anyway provided by Shiva, as I discuss in this answer.)
I should make clear that this story is probably not authentic, but rather a later interpolation to the Shiva Purana, because no story remotely like this is found in any other Puranas or other Hindu scriptures. And there are no other scriptures that mention Vishnu begetting children with women other than his wives. (But there are references to Vishnu having children with Lakshmi in the Sri Sukta of the Rig Veda Khilani, as I discuss in this answer.)
You mentioned the Harivamsa, but story above doesn't occur in the Harivamsa. The Harivamsa does describe, however, a battle between Shiva and Vishnu's incarnation Krishna, in this chapter and this chapter of the Vishnu Parva. (It's also described in this chapter of the Srimad Bhagavatam.)
Krishna's grandson Aniruddha was once kidnapped by the demon Bana (son of Mahabali), who had captured Aniruddha after he discovered that his daughter Usha and Aniruddha were in love. So Krishna came with Balarama and the armies of Dwaraka to defeat Bana and rescue his grandson. They soon defeated Bana's armies, so Shiva and his Gana army came to Bana's aid, since Bana was a great devotee of Shiva. Thus begin an epic battle between Krishna and Shiva, where they launched many powerful Astras (magical weapons) at one another. The battle became so destructive, with Krishna winning, that Bhumidevi the goddess of the Earth worried that the Earth wouldn't survive it, so she asked Brahma to put an end to it. Brahma approached Shiva and asked him to stop fighting:
The lord Brahma looked at Rudra (Shiva) and spoke the following words: "The destruction of great demons has started now (with your consent). Then why are you protecting them? O the one with great arms! I do not like your fighting with Krishna. Also you are not understanding Krishna with your intellect. Your soul is divided in two parts (of which one is Krishna)." When he was advised thus by Brahma, the lord (Shiva), the lord who does not have any decline, entered his (own) body by yoga and visualized the entire three worlds along with all the moving and fixed beings. Then Rudra (Shiva) who is born from a single source, realized his own soul as the soul of Krishna (Brahma - the ultimate soul).
So Shiva realized that Krishna was none other than Vishnu, who is Paramatma the supreme soul and thus undefeatable, so he surrendered. In case you're wondering, Krishna was successful in defeating Bana and rescued Aniruddha, but he spared Bana's life because Bana was the great-grandson of Vishnu's great devotee Prahlada. And Bana too became friendly with Krishna. So Aniruddha married Bana's daughter Usha and everyone went back to Dwaraka.
Strictly speaking neither of the above stories actually describe a battle between Vishnu and Shiva; the first story is about a battle between Vishnu and Shiva's incarnation Vrishabha, and the second story is about a battle between Shiva and Vishnu's incarnation Krishna. As far as I'm aware, the only battle between Vishnu and Shiva themselves is the one described by Parashurama in the Bala Kanda of the Ramayana, and Vishnu won it merely by saying "hum":
Once, all the gods were asking the Grandparent, Brahma, as to who is powerful and who is less powerful among the blue-throated Shiva and Vishnu, but the Grandparent Brahma on inferring the intent of gods started to create adversity among those two, Shiva and Vishnu, for the Grandparent is the best adherer of truthfulness, as truth cannot be demonstrated on hearsay evidence. Owing to their animosity then occurred a fierce and hair-raising war among Shiva and Vishnu, as each aspired victory for himself. By the 'hum' sound of Vishnu that ruinously overpowering longbow of Shiva is broken, and the triple-eyed God, Mahadeva, is frozen.
Then gods along with the assemblages of sages and celestial carana-s have come together and appealed to those two for appeasement in the matter of wielding authority, and then those two superior gods, Shiva and Vishnu, went into a state of amity. On seeing the bow of Shiva rendered inert by the mettlesomeness of Vishnu, from then on the gods along with the assemblages of sages deemed Vishnu to be the paramount.
In case you're wondering, Shiva's bow is the same one that was later broken by Vishnu's incarnation Rama to win Sita's hand in marriage.
On a side note, there is also a story of a battle between Vishnu's incarnation Narasimha, Shiva's incarnation Sharabha, and Vishnu's incarnation Gandhabherunda, which I discuss in this question, but I haven't found a full scriptural basis for that story yet.
It simply means Thou art Brahma(the paramount), Though art Vishnu(the protector), Thou art Rudra(the destroyer/the exterminator), Thou art Prajapathi(the lord of all/ the creator).
There is another mantra which states
Pranoh Brahma, kam brahma and Kham brahma
which is actually one of the definitions for god, which means Brahma(The supreme divine, not to be confused with the creator, 4 headed Brahma) is actually the vital force within every living entity responsible for evolution(pranoh brahma) the prana is brahma, which is also present in a fluid/water form the 'kam' which represents everything the human mind can perceive and understand and also the things beyond this, everything that is real, the number of points (as in the notion from philosophical math) that exist? the past and the present, the logical, the infinite, the vishnu. Finally, the same devine being exists as 'Kham' the ethereal form, beyond everything the human mind can understand and perceive, reality and beyond, the number of points that exist? and also the superset that includes the points that dont exist at all along with all the points that exist, the intuition, the infinite which is both 0/0 and infinity, the shiva.
In other words, they are the same person, the same god, or there is no shiva, no vishnu, just the supreme parabrahma. The fights are just the illusions created in the minds of people, who couldn't understand the complex working of the universe. The human minds perceived Darwinian theory and Mendel's theory to be counter active, it turned out both are true and complementary. The same goes for Shiva and Vishnu.
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