However, investors know there's a difference between a great company and a great stock, and Twitter watchers fretted that enthusiasm may be outpacing fundamentals.

"Twitter the product is one of the most amazing things I've ever seen in 31 years of tech investing," Roger McNamee, Elevation Partners co-founder, said on CNBC's " Fast Money Halftime Report ." "It just came out of nowhere."

In fact, it was the ferocity of the share price increase that scared even those who are bullish on the microblog social media network.

Twitter's debut as a publicly traded stock couldn't have gone much better Thursday, yet market pros remained cautious about its future.

(Read more: Twitter honchos hit $4 billion jackpot)

"It's a trading stock today. Maybe it will be a trading stock for a while," McNamee added. "Me personally, I wouldn't buy here. Then again, I would never buy on the day of any IPO. That's just not how I work. I wait until things settle out and people have figured out what it's really worth. Once the emotions come out of it, that's the time I look to buy a stock like this."

Bob Peck, the first analyst to put a "buy" rating on the stock, told CNBC investors shouldn't be overly impressed with the debut and understand instead that they'll need to be in it for the long haul if they want to get real value.

He spoke as Twitter rolled to a 72.7 percent single-day gain to close at $44.90 after pricing at just $26.

The boom drew its share of skepticism, with Pivotal Research christening the stock with its first "sell" rating, based primarily on valuation. "With a price that pushes into the high 30s and beyond, Twitter is simply too expensive," the firm said.

(Read more: Cashin warns: Nobodyrings a bell at the top)

"We'll see where it shakes out. It's so hard to gauge the true price because it's being driven by supply and demand imbalances," said Peck, analyst with SunTrust. "Anybody buying here is holding for the longer term."

That's based on Peck's view that Twitter will be around $50 by the end of next year, though it will have a hard time getting to the $90 valuation in three years that he said would justify its ownership on a 20 percent return.

One of the differing characteristics of Twitter trading was the accessibility to the retail investor, who is often locked out of IPOs.

Knight Trading logged some 11 million shares traded and UBS added five million.

(Read more: Twitter shines in market debut)

Dave Rovelli, managing director of U.S. equity trading at Canaccord Genuity, said Twitter essentially followed the LinkedIn model, which skyrocketed as well out of the gate by holding back the amount of shares it offered initially.

Still, he said he would be hesitant to hang onto the stock for too long, essentially echoing the call from the Pivotal Research analysis.

"It's going to be all over the map. Eventually they hope it stays up here so they can do a secondary offering at prices up here," Rovelli said. "I wouldn't hold it here, no way. It's way too expensive for me."

—Follow Jeff Cox on Twitter @JeffCoxCNBCcom.

