Portia, as Maya, left, and Rebecca Brooksher, as Tonise in Monster at the Door. Photo Brett Coomer.

The Monster at the Door casts the widest net possible, thematically speaking, pulling in ancient myth and contemporary anxiety, extramarital affairs and end-of-days calamities — not to mention the danger and wonder of art, meteors and the ocean floor.

How playwright Rajiv Joseph pulls it all together — indeed, whether he can — is the mystery and marvel of the rambunctiously ambitious play making its world premiere in the Alley Theatre’s resourceful production.

Joseph has won much praise in the past three years for Bengal Tiger at the Baghdad Zoo, now on Broadway, and Gruesome Playground Injuries, which premiered at the Alley in 2009 and played off-Broadway earlier this year. Once again, he impresses as an original voice, here perhaps a shade too determined to startle, yet maintaining a certain fascination with the play’s ever-shifting mix of the surreal and the ordinary, the grotesque and the blithely comic.

Monster begins simply enough, yet cryptically even there. Maya, an artist, arrives at the headquarters of Connor Fergueson, a global securities firm, to meet with its corporate art curator, Tonise. Maya is one of 20 finalists competing for the $1 million commission to create an artwork for the building’s lobby — but the meeting goes badly. Unconventional Maya distresses the uptight Tonise with her apparent psychic powers, instantly determining that Tonise is having an affair with boss Fergueson, who pops in at one point.

Yet Maya wins the commission and creates her own artistic representation of a giant meteor (the characters keep calling it a mural, but it looks more like a collage or sculpture). Whether because the world already was beginning to unravel, as the meteor-fixated Maya believes, or because her own “meteor” unleashes some uncontrollable force, Maya, Tonise, Fergueson and two other employees, security guard Vince and tax attorney Jesse, find their lives spinning out of control.

To say more would spoil the element of surprise so important to the action — but the characters’ unexpected encounters and strange transformations unfold against periodic reports of offstage earthquakes, plagues and other disasters.

Certain notions keep recurring. That one impossible-to-predict glitch could set the universe unraveling and destroy all life. That a miraculous power to heal must inevitably carry destructive force as well, especially for the wielder of that power. Joseph finds fresh and generally effective ways to dramatize such ideas.

Yet much as one admires the play’s boundless moxie, one sometimes gets the sneaking suspicion Joseph may be including too much. Virtually every element is justifiable for one reason or another, yet the connections tying them together don’t always feel convincing. Though Act 2 has its share of potent scenes, it also has stretches when the play seems to careen out of control as much as the character’s lives.

Nonetheless, imagination and a flair for the bold theatrical stroke can carry a play a long way — especially when its strengths are as well realized as they are in this production.

In her Alley debut, Daniella Topol directs with ingenuity, a sure hand and an understanding of the play’s skewed perspective. With superlative work from designers Kevin Rigdon (sets), Amy Clark (costumes), Tyler Micoleau (lighting) and Jill Duboff (sound), Topol handily realizes some fanciful effects and summons the dreamlike atmosphere required by certain scenes.

Portia (yes, she goes by one name) brings an earthy punch and wry humor to Maya, her declamatory delivery conveying an inborn uniqueness. She impresses as a warily resilient force. Rebecca Brooksher is properly taut and wary as the careful Tonise, handling a drastic midplay change of character with complete conviction.

James A. Stephens gives Fergueson wintry gravitas and toughness, later finding the poignancy in the role’s more human side. Brian Reddy’s Vince is an affable “regular guy” unmoored by strange happenings.

Adam Green’s goofily heroic Jesse cycles through so many phases that it’s almost multiple roles, yet Green somehow spans the entire play with consistency and agreeable humor.

Regardless of whether it all fits together, for most of its running time, The Monster at the Door intrigues — even if it’s just by keeping you guessing what happens next.

The Monster at the Door
7:30 p.m. Tuesdays-Thursdays
8 p.m. Fridays, 2:30 and 8 p.m.
Saturdays, 2:30 and 7:30 p.m.
Sundays, through May 29
Neuhaus Stage, Alley Theatre, 615 Texas
$21-$65