[Query] Are agasthiya's guru hayagreevar another muni and different from Perumal?
There have been such speculations, but they seem to be just speculations, unless proved otherwise. We consider all the 18 purANas as pramANas, which includes brahmANDa purANa, unless you believe the tAmasa purANadi crap. lalitA trishatI, a part of this purANa has been commented upon by Adi Shankara bhagavatpAda and this work is listed among the genuine works of AchArya by Dakshinamnaya Sringeri Mutt. If one looks at the purANa itself:
tatra vAraNashailendraM ekAmranilayaM shivam |
kAmAkShIM kalidoShaghnIM apUjayadathAtmavAn ||
lokahetordayArdrasya chittamAnasya dhImataH |
chirakAlena tapasA toShito.abhUjjanArdanaH ||
hayagrIvAM tanUM kR^itvA sAkShAcchinmAtravigraham |
pUrayan trijagatkR^itsnaM prabhayA dehajAtayA |
prAdurbabhUva purato muneramitatejasaH ||
x x x
eSho.amShabhUto devarShirhayagrIvo mamAparaH |
shrotumicChasi yadyattvaM tatsarvaM vaktumarhati ||
From this, it becomes clear that hayagrIva muni is an avatAra of nArAyaNa. Now, when avatAras like rAma, kR^iShNa etc. are considered as non-different from each other and from Adi nArAyaNa, the same rule should hold good here as well and hayagrIva muni would thus be the same as nArAyaNa or as his avatAra as hayashIrSha.
Here is what shrI bhAskara rAya says in saubhAgya bhAskara:
ashvAnana mahAbuddhe sarvashAstravishArada |
tatastathA jIvito viShNuH hayagrIvo bhUtvA hagagrIvAkhyaM daityaM hatavAn | rahasyajAtamakhilaM devImukhAdeva labdhavAnityAdi | so.ayaM ashvAnano viShNureva tadidaM visheShyamukta vR^ittAntasmAraNena devyanugR^ihItatvAbhiprayagarbham | ata eva mahAbuddhe ityAdi visheShaNadvayaM na stutimAtram ||
shrI bhAsurAnandanAtha here reminds us of the incident from devIbhAgavata mahApurANa [oh yes!] regarding the slaying of the demon named hayagrIva, when viShNu attained hayashIrShatva due to the grace of lalitAmbikA. This also indicates of his direct initiation from bhagavatI mahAtripurasundarI and his direct learning from paradevatA. Moreover, the two adjectives described here, mahAbuddhi and sarvashAstravishArada are not mere stutis but reality, because of his svarUpa as nArAyaNa and because of complete grace of parameshvarI on him.
One should not forget the association of hayagrIva with shrIvidyA. parA tantra speaks of how this school was suited for the first two yugas and not in kali yuga. A few remaining adherents of hayagrIva sampradAya use hayagrIvamahAviShNuH as the R^ishi for trishatI, parA shoDashI, mahApAdukA etc. Moreover, mahApAdukA and guru mantras have the classic hayagrIva bIja embedded. The lineage [Samjokta, correct me if I am wrong] here is: parAshrIH, hayagrIva, agastya, vishvaksena, garuDa, uddhAlaka etc.
vidyAraNya, in his sadAchAra dIpikA [not really endorsed by Sringeri Mutt though as the work of vidyAraNya], speaks of dharmasya prabhurachyutaH and as achyuta revealing dharma to agastya. Agastya, in his short gloss of panchadashI mantra, salutes to his guru hayagrIva as mahAviShNu. It may be noted that the general practice in shrIvidyA is to associate Guru with shiva and not viShNu. There indeed is no rule but that is the general practice.
puNyAnandanAtha, probably different from the author of kAmakalAvilAsa, in his exceptional commentary on the saubhAgya aShTottara of tripurA rahasya, addresses this query when commenting on the name, hayagrIveShTadA. He quotes various pramANas from the Tantra, purANa, Agama etc., and also states the traditional view that the preceptor of kumbhasambhava indeed is viShNumUrti. When we believe that lalitA sahasranAma is of value, that there is hayagrIva way of shrIvidyA, that shrIvidyA has origins in parAmbA etc., it is not really difficult to accept mahaviShNu as agastya's guru, is it? The reason being the pramANa available in these very scriptures. Take it all or take none! The verses in parA tantra, considered of great antiquity by mahAmahopAdhyAya Sri Gopinath Kaviraj, explicitly describes how mahAviShNu descended as hayagrIva R^iShi and initiated agastya into "vaiShNavAchAra" paddhati of shrIvidyA, which was lost in kali yuga. To understand the view of actual practicing lineages, one can refer to medieval works like parA ShoDashI kalpa, hayagrIva tantra [not to be confused with the Buddhist tantra], ratnakosha etc.
Again, these discussions are of no practical use. This is the same as discussing whether Adi Shankara is an incarnation of mahAdeva or not. If you accept them to be avatAras, well and good. If not, read their works and take them based on the works if they are of value. Either case, there is no use debating.
shrIkR^iShNaH sharaNam mama